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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL MEMORANDUM
No. 1218
LECTURE SERIES «BOU]SIDARY LAYER THEORY"
PART n - TURBULENT FLOWS
By H. SchlLchting
Translation of "Vortragsreihe" W.S. 1941/42, Luft-
fahrtforschungsanstalt Hermann Goring, Braunschweig
Washington
April 1949
^^SHY OF FLORIDA
roajMEMTS DEPARTMENT
eAINESV(LLE,FL 32611-7011 udA
-EkCk TM Wo. 1218
TABLE OF CONTENTS
Chapter XIII. GENERAL REMARKS ON TURBULENT FLOWS 1
a. TurlDulent Pipe Flow 1
b. TurlJulent Boundary Layers 3
Chapter XIV. OLDER THEORIES 1^
Chapter XV. MORE RECENT THEORIES; MIXING LENGTHS 11
Chapter XVI. PIPE FLOW I8
a. The Smooth Pipe I8
h. The Rough Pipe 25
Chapter XVII. THE FRICTION DRAG OF THE FLAT PLATS IN LONGITUDINAL
- FLOW '. 30
a. The Smooth Pipe 3I
b. The Rough Pipe J+O
c. The AdmisBihle Roughness l+l
Chapter XVIII. THE TURBULENT FRICTION LAYER IN ACCELERATED,
RETARDED FLOW U3
Chapter XIX. FREE TURBULENCE • I+9
a. General Remarks: Estimations . . . . » h9
h. The Plane Wake Flow 56
c. The Free Jet Boundary 6I
d. The Plane Jet 65
Chapter XX. DETERMINATION OF THE PROFILE DRAG- FROM THE LOSS OF
MDMENTUM ■ . . 66
a. The Method of Betz 69
h. The Method of Jones 72
Chapter XXI. ORIGIN OF TURBULENCE 7I+
a. General Remarks 7^
h. The Method of Small Oscillations 76
c. Results 91
Chapter XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION
LAYER ACCORDING TO THE MEl'HOD 01'" GRUSCHWITZ
(R-EFERENCE 78) 93
a. Integration of the Differential Equation of the Turbulent
Boundary Layer 9j
b. Connection Between the Form Parameters rj and H = i3/5* of
the Boundar-y Layer 96
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MTIONAL AUVTSOEY COMMITTEE FOE AEEOKAliriCS
TECinnCAL MEMOEAMUM NO. 12l8
LECTUEE SERHS "BOIMDAEY LAYER THEOEY"
Part II - Turbulent Flows*
By H. Schlichting
CHAJTEE XIII. GEHERAL REMAEES ON TUKBIOnST FLOWS
a. Turbulent Pipe Flow
The flow laws of the actual flows at high Eeynolds numbers differ
considerably from those of the laminar flows treated In the preceding
part. These actual flows show a special characteristic, denoted as
"turbulence."
The character of a turbulent flow is most easily imderstood in
the case of the pipe flow. Consider the flow through a straight pipe of
circular cross section and with a smooth wall. For laminar flow each
fluid particle moves with uniform velocity along a rectilinear path.
Because of viscosity, the velocity of the particles near the wall is
smaller than that of the particles at the center. In order to maintain
the motion, a pressure decrease is required which, for laminar flow, is
proportional to the first power of the mean flow velocity (compare
chapter I, Part I) . Actually, however, one observes that, for larger
Eeynolds numbers, the pressure drop Increases almost with the square of
the velocity and is very much larger than that given by the Hagen—
Polseuille law. One may conclude that the actual flow is very different
from that of the Polseuille flow.
The following test. Introduced by Eeynolds, . is very instructive:
If one inserts into the flowing fluid a colored filament one can observe,
for small Eeynolds numbers, that the colored filament Is maintained down-
stream as a sharply defined thread. One may conclude that the fluid
actually flows as required by the theory of laminar flow: a gliding
along, side by side, of the adjoining layers without mutual mixing
(laminar = layer flow). For large Eeynolds numbers, on the other hand,
one can observe the colored filament, even at a small distance downstream
from the inlet, distributed over the entire cross section, that Is, mixed
*"Tortragsrelhe 'Grenzschichttheorle. ' Tell B: Turbulente Stromungen."
Zentrale fiir wlssenscliaftliches Berichtswesen der Luftfahrtforschimg des
Goneralluftzeugmeisters (ZWB) Berlln-rAdlershof , pp. 15^+— 279. The
original language version of this report is divided Into two main parts.
Tell A and Tell B, which have been translated as separate WACA Technical
Memorandums, Nos. 1217 and 12l8, designated Part I and Part II, respectively.
This report is a continuation of the lecture series presented in part I, the
equations, figures, and tables being numbered in sequence from the first
part of the report. For general information on the series, reference
should be made to the preface and the Introduction of Part I.
2 NACA TM no. 12l8
to a great extent vlth tlie rest of the fluid. Thus the flow character
lias changed completely for large Eeynolds numbers: A pronounced
transverse mixing of adjacent layers takes place. Irregular additional
velocities in the longitudinal and transverse directions are superposed
on the main velocity. This state of flow is called turhulant. As a
consequence of the mixing the velocity is distributed over the cross
section more uniformly for turhulent than for laminar flow (compare fig. 17,
part -I) . For turhulent flow there exists a very steep velocity increase
in the immediate neighhorhood of the wall and almost constant velocity in
the central regions. Consequently the wall shearing stress is considerably
larger for turhulent than for laminar flow; the same applies to the drag.
This follows also from the fact that in turbulent flow a considerable
part of the energy is used up in maintaining the turbulent mixing motion.
The exact analysis of a turbulent flow shows that at a point fixed
in space the velocity is subjected to strong irregular fluctuations with
time (fig. 72). If one measures the variation with time of a velocity
component at a fixed point in space, one obtains, qualitatively, a
variation as shown in figure 72. The flow is steady only on the average
and may be interpreted as composed of a temporal mean value on which the
irregular fluctuation velocities are superimposed.
The first extensive experimental Investigations were carried out
by Darcy (reference 60) in connection with the preliminary work for a
large water-distributing system for the city of Paris. The first quanti-
tative experiments concerning laminar pipe flow were made by Hagen
(reference 95)- The first systematic tests regarding the transition from
laminar to turbulent flow were made by Osborn Eeynolds (reference 6l) .
He deteimlned by experiment the connection between flow volume and pressure
drop for turbulent flow and investigated very thoroughly the transition
of the laminar to the turbulent form of flow. He found, in tests of
various velocities and in pipes of various diameters, that transition
always occurred at the same value of the Eeynolds number: — . This
Eeynolds number is called the critical Eeynolds number. The measurements
gave for the pipe flow:
Ke.^i^ = (^) = 2300 (13.1)
crit v^ y /crit
For Ee < Ee^^-j^^ the flow is laminar, for Ee > I^e^-^j^-j., turbulent. Later
on it was ascertained tliat the numerical value of Ee^^-j^j^^ is, moreover,
very dependent on the particular test conditions. If the entering flow
was very free of disturbances, laminar flow could be maintained up to
Ee = 24000. However, of main interest for the technical applications
is tlio lowest critical Eeynolds number existing for an arbitrary disturbance
of tlie entering flow, due either to irregularities in the approaching flow
or to vortices forming at the pipe inlet. Concerning the drag law of the
pipe Eeynolds foimd tliat the pressure drop is proportional to the 1.73
power of tlie mean flow velocity:
-1-73
A"D ---. u
NACA TM Wo. 1218
Id. Turtulent Boundary Layers
Becently it was determined that the flow along the surface of a body
(■boimdary layer flow) also can be turbulent. We had found, for instance,
for the flat plate in longitudinal flow that the drag for laminar flow is
proportional to \J '^q (compare ecjuation (9.I8), Part I.) However, towing
tests on plates for large Reynolds numbers carried out by Froude
1 B'l
(reference 96) resulted in a drag law according to which W ~ Uq ' ' . More-
over, the drag coefficients in these measurements remained considerably
higher than the drag coefficient of the laminar plate flow according to
equation (9.I9), Part I. Presumably this deviation is caused by the
turbulence of the boundary layer.
A clear decision about the turbulent flow in the boundary layer was
obtained by the classical experiments of Eiffel and Prandtl concerning
the drag of spheres in 19114. (reference 62). These tests gave the following
results regarding the drag of spheres (compare fig. 73). The curve of
drag against velocity shows a sudden drop at a definite velocity V^pit.*
although it rises again with further increasing velocity. If one plots
the drag coefficient c^^ = w/f ^ U^ (F = frontal area) against the
Reynolds number Uod/v, c^ shows a decrease to 2/5 of its original value
at a definite Reynolds number (^®crit^' ^s^^L'tl explained this phenomenon
in 1911j-. He was able to show that this drag decrease stems from the laminar
boiindary layer changing to turbulent ahead of the separation point. The
resulting considerable rearward shift of the separation point causes a
reduction of the vortex region (dead water) behind the sphere (fig. 7k).
This hypothesis could be confirmed by experiment: by putting a wire ring
on the sphere (sphere diameter 28 centimeters, wire diameter 1 millimeter)
one could attain the smaller drag at smaller V^^^^j^^ and Pe^j,^.^^. The wire
ring is put on slightly ahead of the laminar separation point; it causes a
vortex formation in the boundary layer, which is thus made turbulent ahead
of the separation point and separates only farther toward the rear. By
means of the wire ring the boundary layer is, bo— to— say, "infected" with
turbulence. Due to the mixing motions which continually lead high velocity
air masses from the outside to the wall, the turbulent boundary layer is
able to overcome, without separation, a larger pressure increase than the
laminar boundary layer.
The turbulence of the friction layer is of great Importance for all
flows along solid walls with pressure increase (diffuser, wing suction
side). It is, however, also present in the flow along a flat plate where
the pressure gradient is zero. There the flow in the boundary layer is
laminar toward the front, experiencing transition to the turbulent state
further downstream. Whereas the laminar boundary layer thickness increases
downstream with xl/^^ the turbulent boundary layer thickness increases
k WACA TM No. 1218
approximately as x ' ; that is, for the t'orb-olent bo'ondary layer the
increase of the houndary layer thickness is considerably larger, (fig. 75).
The position of the transition point x ., is given by (fig. 75):
(13.2)
In comparing the critical Eeynolds numbers for the pipe and the plate
one must select r and S, respectively, as reference lengths. The
equation for the flat plate is, according to Blasius (reference 8)
(compare equation (9.21a))
or
U^5 _ . U^x
■'o
y (13.3)
n
Thus, with (— ) = 25000 :
^ ^ ^crit
(^) = 5.500 = 2500 (flat plate) (13.^)
^ -^crit
^max ^
This critical Eeynolds number must be compared with — at the
transition point for the pipe. Due to the parabolic laminar velocity
distribution in the pipe ri ^ = 2 u, and because r = i d, then, for
the pipe, u r/v = ud/v. According to equation (I3.I), (•^) = 23OO
crit
for the pipe. Thus the comparable critical Reynolds numbers for pipe and
plate show rather good agreement.
CHAPTER XrV. OLDER THEORIES
The first efforts toward theoretical calculation of the turbulent
flows go back to Reynolds. One distinguishes in the theory of turbulence
two main problems:
1. The flow laws of the developed turbulent flow:
The space and time velocity variations affect the time average of
the velocity; they act like an additional internal friction. The problem
is to calculate the local distribution of the time average of the velocity
components, and thus to gain further information concerning, for instance,
the friction drag.
NACA TM No. I2l8
2. Origin' of turbulence:
One Investigates under what conditions a small disturbance,
superposed on a laminar flow, increases with time. According to whether
or not the disturbance ir. "reases with time, the laminar flow is called
unstable or stable. The :':vestigation in question is therefore a stability
investigation, made to clarify theoretically the laminar— turbulent transi-
tion. These investigations aim particularly at the theoretical calculation
of the critical Eeynolds number. They are, in general, mathematically
rather complicated.
The first problem, since it is the more important one for general
flow problems will be our main concern. The second will be discussed
briefly at the end of the lecture series.
As to the first problem, that of calculation of the developed
turbulent flow, one may remark quite generally that a comprehensive
theoretical treatment, as exists for laminar flow. Is not yet possible.
The present theory of the developed turbulent flow must be denoted as
semi— empirical. It obtains its foundations to a great extent from
experiment, works largely with the laws of mechanical similarity, and
always contains several or at least one empirical constant. Nevertheless
the theory has contributed much toward correlating the voluminous experi-
mental data and also has yielded more than one new concept.
For the numerical treatment one divides the turbulent flow, unsteady
in space and time, into mean values and fluctuation quantities. The mean
value may a priori be formed with respect to either space or time. We
prefer, however, the time average at a fixed point in space, and form
such mean values of the velocity, pressure, shearing stress, etc. In
forming the mean values one must not neglect to take them over a suffi-
ciently long time interval T so that the mean value will be independent
of T. Let the velocity vector with its three mutually perpendicular
components be
•»v = iy + Jv + kw (1^-1)
For a tijrbulent flow the velocity components are therefore functions of
the three space coordinates and the time:
u = u (x, y, z, t)
V = V (x, y, z, t) (li+.2)
w = w (x, y, z, t)
The time average for the component u, for instance, is formed as follows:
(li^.3)
*nTroughout the text, underscored letters are used in place of
corresponding German script letters used in the original text.
WACA TM No. 1218
If u, V, w are independent of t^ and T, the motion is called steady
on the average J or quasisteady. A steady turbulent flow^ in the sense
that the velocity at a point fixed in space is perfectly constant, does
not exist. The velocity fluctuations are then defined by the equations
u = u + u'
T = V + V
w = w + w
(ih.k)
and in the same way for the pressure:
P + P'
(1^.5)
The time average of the fluctuation quantities equals zero, according to
definition, as the following consideration will show immediately:
to+T
,to+T
u'dt = -
Ito+T
udt-^
dt = u - u = {Ik. 6)
ThuE
u-
(ik.V
The Additional "Apparent" Turbulent Stresses
As a result of the velocity fluctuations additional stresses
( = apparent friction) originate in the turb'olent flow. This is readily
illustrated for instance by the case of the simple shearing flow u = u(y)
(fig. 76). Here ~ = 0; however, a fluctuation velocity v' in the
transverse direction is present. The latter causes a momentum transfer
between the adjoining layers across the main flow. This momentum transfer
acts like an additional shearing stress t. Whereas in laminar flow the
friction is brought about by the molecular momentum exchange, the turbulent
exchange of momentum is a macroscopic motion of, mostly, much stronger effect.
The equations of motion of ths turbulent flow, with this tiirbulent
appa.rent friction taken Into consideration, can be obtained from the
Navler— Stokes differential equations by substituting equation (lli.lj-) into
the latter and then forming the timc^ averages in the Wavier— Stokes
ilfferentlal equations. To that purpose the Navier— Stokes differential
equations (3.I6) are written in the form:
NACA TM No. 12l8
'Sa ^(u^) S(uv) S(uw)
p|^ " ~ax
az
P T— + — ^ + + — ^ -
[at Bx Sy hz
St Bx Sy Sz
ap /S^u B^u sV
^ + ^^ ( -2 + -^ + — 2
\Sx Sy Sz /
Sp /S2t B2v S2v^
— + n' — - + +
\Sx Sy Sz >
\Sx Sy Sz
-— + — + — =
ox Sy Bz
> (1^.8)
By introducing equations (l^A) and (lli-.5) and forming the time averages
one first ottains from the continuity equation:
5u Sv c)w _
Sx Sy Sz
(11..9)
and thus also:
Su' St' Sw'
+ + =
Sx Sy
(11^.10)-
By introduction of equations (lii.U) into the left side of equation (14.8)
one obtains expressions asj for instance,
2 2 2 — 2
u = (u + u' ) = u + 2 u u' + u' etc.
In the subsequent formation of the time average the squared terras in
the "barred quantities remain unchanged since they are already constant
with respect to time. The mixed terms^ as for Instance u u', . . .
and also the terms that are linear in the fluctuation quantities are
eliminated in forming the average "because of equation (1^.7). However,
2
the terms that are quadratic in the fluctuation quantities as u' ,
a*v*, . . . remain. Thus one o"btains from the equation system
(equation (ll4-.8))j after forming the time average the following system
of equations:
8
KACA TM No. 1218
V dx 'dj 3z
/_5t -Bv — Sv
Sx 5y Sz
— + uAu — p <
Sx hj Sz
^ + MAT -p p^^ + ^ll- + ^'^'
Sy Sx ^ Sz
/_^ _^ -5w 1 Bp .-
plu^- + V-— + w— 1= - -ii + pAw -p
\ ox oj oz / oz
2
I Sx Sy Sz
^
> (14.11)
The left side now formally agrees with the Navier-Stokes differential
equations for steady flow if one writes instead of u, y, w the time
averages of these quantities. On the right side additional terms which
arise from the fluctuations have been added to the pressure and friction
terms .
Remembering that in deriving the Wavier— Stokes differential equations
one could write the resultant surface force per unit volume "by means of
the components of a stress tensor according to equation (3.7) in the form
5x Sy
St Sa Bt
,jl^Z^ y. yz
Sx
k \ + — ^— +
Sy Sz
-^ Bx
^
Bz>
>
ilk.l2)
one recognizes "by comparison with (equation (lU.ll)) that one may introduce
for the quantities added by the fluctuation motion a syiimietrical stress
tensor in the following manner:
,2
a = — pa'
X ^
T = — pu'v'
xy
V " " ^'^'^'
ay = - pv'2
T = — pu'w'
XZ ^
yz
pv^^'ir
tTJ«
T = — pu'w'
XZ
Ty^ = - PV'W'
a^ = - pw»2
^
J
(14.13)
One has therefore^, for the mean values of the quaslsteady flow, the
following equations of motion:
NACA TM No. 12l8
, -Su -Su -Su
PJU— + T— + W—
3X aj oz
/-Civ' _Sv -Sv
P(U + T + W
Sx Sy Sz
Ai^ + v^ + w^V
Bx Sy
Sp ^- ^^x
ox dx
St
Bt
xy
Sy
St
Sp A— " xy
By Sx By
Bp + ^v + -^ + -J-
Bz Bx By
JLZ
xz
Bz
Bt
+
yz
Ba.
z
Bz
> (1^.1^)
y
The continuity equation (equation (li<.,9)) also enters. The boundary
conditions are the same as for laminar flow: adhering of the fluid to the
wall, that is, on the solid walls all velocity components equal zero.
According to equation (l^.lif) the mean valuep of the turbulent flow obey
the same equations of motion as the velocity component's of a laminar flow,
with the friction forces, however, increased by the apparent stresses of
the turbulent fluctuation motion. But since the fluctuation velocities
u*, v', . . . and particularly their space distribution are unknown,
equations (lij-.lij-) and (l4.13) are, at first, rather useless for tlie
calculation of a turbulent flow.
Only when one will have succeeded in expressing the fluctuation
quantities "'
u' ~, u'v', ... in a suitable manner by the time averages
will it be possible to use equations (l4.lU) to calculate.
in particular, the mean values u, v, w.
A first expression of this irind which brought however little saccess
was originated by Boussinesq (reference 6k). He introduced, aside from
the ordinary viscosity coefficient, a new viscosity coefficient of the
apparent turbulent friction. In analogy to the stress tensor for
laminar flow which is, according to equation (3.I3):
cr T T
X 'xy xz
''xy ^y ""yz
""xz "^jz ^z
p
— —
p
p
+ u-
Bu Bu Bu
Bx By Bz
Bv Bv Bv
Bx By Bz
Bw Bw Bw
Bx ^y Bz
+ 1^
Bu Bv
Bw
Bx
Bx
Bx
Bu
By
Bv Bw
By By
Bu
Bz
Bv
Bz
Bw
Bz
^^^ (14.15)
Boussinesq puts for the apparent turbulent friction:
Or. Bu ^ _ „ / Bu , Bv
X dx xy \$,j 3z
(1I+.16)
Then there corresponds to the laminar viscosity coefficient n the mixing
factor pg;
10 NACA TM No. 12l8
H~pe or v~e
The kinematic viscosity of the turhulent flow (apparent friction) is
usually very much larger than that for the ordinary laminar friction.
(Hundred— or thounsandfold or more). In general, one may therefore
altogether neglect the ordinary viscosity terms mAu, ... in
equation (lij-.li<-). Only at the solid walls where due to the no— slip
condition
u=v=w=0 as well as u* = v' = w' =0
the apparent turbulent friction disappears, does the laminar friction
again hecome dominant. Thus there exists in every turbulent friction
layer in the immediate neighborhood of the wall a very narrow zone where
the flow is laminar. The thickness of this laminar sublayer is only a
small fraction of the turbulent boundary layer thickness.
One can easily understand from the example of the simple shear flow
according to figure 76 that in a turbulent flow the mean value u'v» is
different from zero. For this case, a correlation exists between the
fluctuation velocities u' and v' in the following manner: The
particles with negative v' have "mostly" a positive u*, since they
come from a region of larger mean velocity u.* The parts with positive
v', on the other hand, have "mostly" a negative u', because they come
from a region with smaller u, and retain in the transverse motion
approximately the x-momentum of the layer from which t hey come. Thus,
"mostly" u'v' < and, therefore, the time average u'v* < 0. Therefore,
the shearing stress is this flow is:
= - p u'v' >
xy
In measuring turbulent flows one usually measiH'es only the mean
values u, V, . . . since only they are of practical interest. However,
in order to obtain deeper insight into the mechanism of the turbulent
flow, the fluctuation quantities have recently been measured and also
their mean squares and products:
V^.2 v^ v^
According to measurements by Eelchardt (reference 65) in a rectangular
tunnel (width 1 millimeter, height 2k centimeters) the maximum value
^"Mostly" is to indicate that particles with different signs,
thoui^Ji not excluded, are in the minority.
NACA TM No. 12l8
11
\[^'-
of V li* J for instance, equals 0.1 3 vl^, the majcimum -value of
equals 0.05 u^^^. Both naxima lie in the neighborhood of the wall.
One may saj, therefore, that in this case the turhulence is strongest
near the wall.
In a flow that is homogeneous (wind t'onnel), turbulent fluctuations
are also always present to a varying degree. They determine the so— called
degree of turbulence of a wind tunnel. Since the measurement of the
fluctuation quantities is rather difficult (hot— wire method), a more ■
convenient measuring method has been chosen, for the present, for determin-
ation of the degree of turbulence of a wind tunnel: namely, the determin-
ation of the critical Eeynolds number for the sphere from force measure-
ments or pressure distribution measurements. One defines as critical
Reynolds number the one where the drag coefficient c^ = 0.3. It
becomes clear that a unique connection exists between the critical Reynolds
number and the turbulent fluctuation velocity in the sense that the
critical Reynolds number of the sphere is the lower, the higher the
turbulent fluctuation velocity. According to American measorements
(reference 97) the connection between the longitudinal fluctuation and
the measured critical Reynolds number for the sphere is as shown in the
following table:
i.'/s
O.OOij-
0.0075
0.012
0.017
0.026
«%rlt ^°"^
2.8
2.k
2.0
1.6
1.2
In addition to the apparent Increase of viscosity, the turbulent
fluctuation motion has other effects: It tends to even out any tempera-
ture differences or variations in concentration existing in a flow.
The diffusion of heat, for instance, is much larger than for laminar flow,
because of the exchange motions which are much stronger in turbulent
flow. A close connnection therefore exists, for Instance, between the
laws of flow and of heat transfer from a heated body to the fluid flowing
■by.
Ninth Lecture (February 2, 19i<-2)
CHAPTER XV". MORE RECENT THEORIES j MIXING LENGTH
In order to make possible a quantitative calculation of turbulent
flows, it is necessary to transform the expressions for the apparent
turbulent stresses (equation (ll4-,13)) in such a manner that they no longer
contain the unknown fluctuation velocities but contain the components of
the mean velocities. Consider, for that purpose, a particularly simple
12 MCA TM No. 12l8
flow, namely a plane flow which has the same direction everywhere and a
■velocity varying only on the different stream, lines. The main— flow
direction coincides with the z— direction; then:
u = il(y) V = w = (15.1)
Of the shearing stresses, only the component t = t" ia present, for
which from equation (114-.13) as well as from Boussinesq's equation,
equation (llf. 15). there resiilts:
du
T = - p u'v» = p e ii^ (15.2)
dy
This formula shows that | t |/p equals the square of a velocity. One
putSj therefore, for use in later calculations.
u»v'
(15.3)
and denotes v.,,. as shearing stress velocity. Thus this shearing stress
velocity is a measure of the momentum transfer "by the turhulent
fluctuation motion.
According to Prandtl (reference 66), one may picture the turhulent
flow mechanism, particularly turhulent mixing, in the following simplified
manner: Fluid particles, each possessing a particular motion, originate
in the turbulent flow; they move for a certain distance as coherent masses
maintaining their velocity (momentum). One now ass-jmes that such a fluid
particle which originates in the layer (y, — l) and has the velocity
u(y-| — l) moves a distance I = mixing length normal to the flow (fig. 77)'
If this fluid particle maintains its original velocity In the x— direction
it will have, in its new location y^, a smaller velocity than Its new
surroundings, the velocity difference heing
-u' = uij^) - u(y-^ - I) with v' >
Likewise a fluid particle coming from the layer (y-, + l) to y-|^ has
at the new location a greater velocity than the surroundings there; the
difference Is
NACA TM No. 12l8 13
Tij and u^ give the turbulent velocity fluctuation in the layer y, .
One ohtalns for the mean value of this velocity fluctuation
|u'| = i (|u^|+ lu^l) = I
(t)l
(I5.i^)
From thla equation one ohtains the following physical interpretation for
the mixing length Z:
The mixing length signifies the distance in the transverse direction
which a fluid particle must travel at the mean velocity of its original
layer so that the difference "between its velocity and the velocity of the
new location equals the mean velocity fluctuation of the turhulent flow.
It is left open whether the fluid particles in their transverse motion
fully maintain the velocity of their original layer, or whether they have
partly assumed the velocity of the traverse layer and then travelled
larger distances in the transverse direction. The Prandtl mixing length
which is thereby introduced has a certain analogy to the mean free path
of the kinetic theory of gases, with, however, the difference that there
one deals with microscopic motions of the molecules, here with macroscopic
motions of larger fluid particles.
One may picture the origin of the transverse fluctuation velocity v'
in the following way:
Two fluid particles flowing from the layers (y-, + l) and (y-, — Z)
meet in the layer y-[_ in such a manner thab one lies behind the other:
the faster (y-|_ + Z) behind the slower (y-i — 2). They then collide
with the velocity 2u' and glveway laterally. Thereby originates the
transverse velocity V , directed away from the layer y-, to both sides.
If, conversely, the slower of the two particles is behind the faster,
they withdraw from each other with the velocity 2u' . In this case the
space formed between them is filled up out of the surroundings. Thus
originates a transverse velocity v* directed toward the layer y-,.
One concludes from this consideration that v' and u' are of the same
order of magnitude and puts
|v'| = number |uM = ntunber Z — (I5.5)
In order to express the shearin g str ess according to equation (I5.2) one
has to consider the mean value u*v* more closely. The following
conclusions can be drawn from the previous considerations.
Ik
NACA TM No. 12l8
The particles arrlTing in the layer y-, with positive t* (from
"below, fig. 77) have "mostly" a negative u' so that u'v' is negative.
For the particles arriving with negative v'j u* is "mostly" positive,
so that u'v' is again negative. "Mostly" signifies that particles with
different sign are not wholly excluded, but are strongly outnumbered.
The mean value u'v» is therefore different from zero and negative. Thus
one puts
u»v« = - k lu'l Iv'l
(15.6)
with k^O; 0<k<l. The niimerical factor k, also called the
correlation coefficient is not known more closely. According to
equation (I5.5) ^^^ (15»^) one now obtains
u«v' =
r,\2
- number Z^/duV
(15.7)
the "number" in this equation being different from the one in
equation (15.5)* If on.e includes the "number" in the unknown mixing
length, one can also write
^TTTT = _ 22/duJ
(15.8)
and thus finally obtains for the turbulent shearing stress according to
equation (I5.2)
P 2
2/du\2
Considering that the sign of t also must change with the sign of
it is more correct to write
du
T=pi2
du
iy
du Prandtl's
^7 formula
(15.9)
This Is the famous Prandtl mixing length formula which has been very
successful for the calculation of turbulent flows.
If one compares this formula (equation (15.9)) with the equations
of Boussinesq where one had put t = e -r— ( e = mixing factor = turbulent
analogue of the laminar viscosity \x) , one has for the mixing factor
RACA TM Ro« 12l8 15
(15.10)
c = p2^
du
The turlaulent mixing factor e Is in moat cases larger than the laminar
■viscosity [x "by several powers of ten. Moreover the mixing factor e
is dependent on the velocity and on the location and tends toward zero
near a wallj "because there the mixing length goes toward zero.
If one compares Prandtl's formula equation (I5.9) with Boussinesq«s
equation (15.2) one could f)erhap8 think at first that not much has been
gainedj since the unknown quantity e ( = apparent viscosity) has been
replaced by the new unknown I = mixing length. Nevertheless Prandtl's
formula is considerably better than the old formula for the following
reason: It is known from tests that the drag for turbulent flow is
proportional to the square of the velocity. According to equation (15.9)
one obtains this square law for drag by assuming the mixing length to be
independent of the velocity, that is, by assuming the mixing length to
be purely a function of position. It is considerably easier to make a
plausible assumption for the length I = mixing length than for the
apparent turbulent viscosity e, and therein lies the considerable
superiority of Prandtl's formula equation (I5.9) over Boussinesq's
equation (I5.2).
In many cases the length 2 can be brought into a simple relation
to the characteristic lengths of the respective flows. For the flow along
a smooth wall I must, at the wall itself, equal zero, since all trans-
verse motions are prevented at the wall. For the flow along a rough wall,
however, the limiting value of I at the wall equals a length of the
order of magnitude of the height of the roughness.
It would be very useful to have a formula permitting the determina-
tion of the dependence of the mixing length on the position for any
arbitrary flow. Such an attempt has been made by v. Karman (reference 68).
V. Karman makes the assumption that the inner mechanism of the turbulent
flow is such that the motion at various points differs only with respect
to time— and length-scale, but is otherwise similar (similarity hypothesis).
Instead of the units of time and length one may select those of velocity
and length. The velocity unit that is Important for the turbulent motion
is the shearing stress velocity v^ according to equation (I5.3). The
corresponding unit of length is the mixing length Z.
In order to find the quantity I from the data of the basic flow
u(y), V, Karman applies the Taylor development* for u(y) in the neigh-
borhood of the point y-, .
u(y) = u(y^) + (y - y^) ^0) ^ir('^- ^1^^ (^) + ^15.11)
* In the following, the bar over the mean velocity will be omitted,
for simplification.
16
NACA TM No. 12l8
The length 2 cannot depend on the velocity ^.(y-,), since according
to Newton's principle of relativity the addition of a constant velocity
has no influence on the course of motion. Thus
■du\
and the higher derivatives remain as characteristic data of the "basic
flow. The simplest length to "be formed from it is
du
dy
d!u
dy^
V. Earman puts therefore
(15.12)
According to this formula I is not dependent on the amount of velocity
"but only on the velocity dlstrl"butlon. Thus 2 is a pure position
function as required ahove. In equation (15.I2) k, is an empirical
constant which must "be determined from the experiment. To arrive further
at the tur"bulent shearing stress, v. Kannan also maintains Prandtl's
equation (15.9).
In generalizing equation (I5.9) one o"btains, according to Prandtl,
the complete expression for the tur"bulent stress tensor of a plane flow
in the form
/CT^ T,
\:^ %
\= pi'
du
dy
Sx
Su I ^
Si
^
(15.13)
du Sv
^ dx '" Sy
2 ^
The common factor on the right side signifies the tur"bulent mixing
factor according to equation (15.IO).
NACA TM No. 12l8 I7
Tenth Le6ture (February 9, 19k2)
Flow Along a Smooth Wall
¥e will immediately make a first application of Prandtl's formula
(equation (15.9)) for the flow along a smooth wall. The normal distance
from the wall is denoted as y. Let the wall coincide with the x-axis.
For the Telocity distribution then, u = u(y). For this case one sets
the mixing length in the neighborhood of the wall proportional to the
distance from the wall
I = K.J ' (15.1^)
the constant k must he determined from the experiment. Moreover one
makes the assumption that the shearing stresb t is constant in the
entire flow region; then the shearing stress Telocity v^ according to
equation (15.3) also is constant. If one further neglects the laminar
friction^ one obtains from equations (I5.2), . (I5.9), (15.14)
or
^ = I*
dy Kj
and by integration
u = — 2n y + constant (I5.I5)
In determining the constant of integration one must pay attention to the
fact that the turbulent law equation (I5.9) does not apply right up to
the wall but that Tery near to the wall an extremely thin laminar layer
is present. From the laminar Tlscoslty \x and the turbulent shearing
stress Telocity t^ one can form the length v/t^. The constant of
Integration in equation (I5.I5) is determined from the condition that
u = for y = yQ. Thus there results, according to equation (I5.I5)
u = ^ (2n y - Zn y^^) (15-16)
The as yet unknown distance from the wall yQ is set proportional to
the length v/t^, thus
18 NACA TM No. 12l8
yo = P =fe - (15.17)
where 3 signifies a dimensionless constant. Thus one finally ottains
for the velocitj disbritution at the smooth wall
T^
y^*
u = ^ (in ±^ - in ^) (15.18)
that is, a logarithmic Telocity distribution law. It contains two
empirical constants k. and p. According to measurements «; = O.k.
From equation (I5.I8) one can see that the dimensionless velocity u/v>. = cp
can he represented as a function of the dimensionless distance from the
wall T\ = v^(.y/v. The latter is a sort of Eeynolds number, formed with the
distance from the wall y and the shearing stress velocity v^.. Thus one
obtains for larger Reynolds numbers from equation (I5.I8) the following
universal velocity distribution law
Cp(Ti) = A In T\ + B (15.19)
with A = l/n = 2.5. For smaller Eeynolds numbers, where the laminar
friction also has a certain influence, tests gave the velocity distribution
larf
Cp(Tl) = C Tl'' (15.20)
or
t - ^ (^Y (15.20a,
with the exponent n equalling about l/?. These universal velocity dis-
tribution laws according to measurements for pipe flow are given in
figure 78. They will be discussed in more detail in the following chapter.
CHAPTER rVI. PIPE FLOW
a. The Smooth Pipe
Among the various turbulent flows of practical importance, pipe
flow was investigated with particular thoroughness because of its great
practical importance. We shall therefore consider the pipe flow first.
It will be noted at this point that the flow laws of the pipe flow may
be applied to other cases, as for instance the plane plate in longituainal
flow. Consider a straight pipe of circular cross section and with a smooth
wall. Let y be the radial coordinate measured from the pipe axis. The
WACA TM No. 1218 19
"balance of forces "between the shearing stress t and the pressiire
drop P-, — Pp on a piece of pipe of length L yields as "before for the
laminar flow according to equation (2..1a)j the relation:
-^-^1 (16.1)
This formula applies equally to laminar and turbulent flow. In it t
now signifies the sum of the laminar shearing stresses and of the apparent
turhulent shearing stresses. Over a cross section, t is proportional
to y. The shearing stress at the wall Tq may "be determined
experimentally "by measurement of the pressure drop:
Pn - P2 r
r^=-^—^L . (16.2)
For the tur'bulent flow the connection "between pressure drop and flow
volume Q = irr u must "be obtained from tests.* In the literature there
exists a very great number of pipe resistance formulas. Only those serve our
purpose which satisfy Eeynolds' law of similarity. One of them is the
formula of H. Blasius (reference 69), set up partic-ularly carefully,
which is valid for a smooth wall and for Eeynolds numbers
Ee = ud/v^ 100 000.
If one introduces, as "before in equation (2.6), the dimensionless
pipe resistance coefficient X "by the equation
!Ll^ = iPu2 (16.3)
L d 2 .
X is, according to Blasius:
,- ,x-iA
X = 0.316U(^^J (l6.k)
Comparing equations (l6.2) and (I6.3) one finds:
^o = I P ^^ ■ (16-5)
and therefore according to equation (iG.k):
_7/k lA -lA
Tq = 0.03955 p a'^ V ' d (16.6)
*In the following, u is, for the pipe flow, the mean flow velocity
at the cross section, as distinguished from the time average in the previous
sections.
20 NACA TM No. 12l8
If one introducsBj moreover, instead of the diameter d the radius r,
the nximerical factor in this linear equation must be divided by
1/4
2 ' =1.19. Thus T becomes:
o
^o = 0.03325 P u^/^ V^A r-lA = p v^2 (ig^^)
where the shearing stress velocity is defined by the wall shearing stress:
v^ = ^ (16.8)
If one finally factors the quantity v^,. in equation (16.7) into v^^. ' x
v^ ' , one obtains:
This equation is very similar to equation (15.20a); however, the mean
velocity now takes the place of the local velocity and the pipe radius
takes the place of the distance from the wall. One passes first from
the mean velocity to the maximum velocity u-|_; based on measurements of
Wikuradse (reference 70) u = 0.8 u,, and therewith follows from
equation (16.9):
u, f^*r\^/^
If this formula is assumed to be valid for any distance from the wall,
one obtains:
^^ , B.lk (^-ff (16.10)
or
<p = 8.1k y^l'^ (16.11)
This is the so— called 1/7— power law for the velocity distribution; its
form was already given in equation (15.20a). The coefficients n and t].
NACA TM No. 12l8 21
still uiLlaaown there^ have now "been determined on the tasis of the
resistance law of the pipe flow. Figure 78 shows, according to measure-
ments of Nikuradse (reference 70) that this law is well satisfied in the
range of Reynolds numhers up to 100,000. Naturally this law of velocity
distribution can apply only to the region of Reynolds n^jimbers where the
pipe resistance law given by equation {l6.h) is valid, since it was
derived from this law.
For pirrposes of later calculations we shall derive from
equation (16.IO) the shearing stress velocity v^. One obtains:
7/8. V 1/8
v^ = 0.150 u (^] (16.12)
7 /fi 1
with 8.7^ = 6.65 and = O.I5O. From equation (16.I2) follows:
6.65
Tq = p rf = 0.0225Pu'^/^ (tT^^ (I6.I3)
\y/
This relation will be needed later.
Comparing measured velocity distributions with equation (16.IO)
one can state that outside of the range of validity of equation (16.IO),
namely for Re > 100,000, a better approximation is obtained by the
power 1/8, l/9j or l/lO instead of l/7- The measurements concerning
the pipe resistance (fig. 8I) show an upward deviation from the formula
of Blasius for larger Reynolds numbers.
The logarithmic velocity distribution law, equation (15.19)^ derived
in the previous chapter has been verified by Nilniradse (reference 70) on
the basis of his meas'jrements for the smooth pipe. For this purpose from
the measured press'ore drop for each velocity profile one first determines
the wall shearing stress according to equation (l6.2) and from tha t,
according to equation (16.8), the shearing stress velocity v^ = \/t /p.
Then the dimensionless velocity cp = u/v^ can be plotted against the
dimensionless distance from the wall t] = yv^(./v. The measurements of
Nikuradse in a very large range of Reynolds numbers. He = k X 10^ up to
32^4-0 X 103, lie very accurately on a straight line if one plots cp against
log r] (fig. 78). The straight line has the equation:
cp = 2.5InTi + 5.5
(l6.11f)
This gives, by comparison with equation (I5.I8), the following nimerical
values for the coefficients K and P
22 NACA TM No. 12l8
K = oAoo p = 0.111 (16.15)
Mixing Length
From the measured Telocity distribution and the measirred pressure
drop the distribution of the mixing length over the pipe cross section
can he determined according to equations (l6.2)j (I6.I), and (15.9).
T = Tq — (y = distance from the pipe axis). This determination of the
mixing length from the meaS'jrements in the pipe was made hy Wikuradse
(reference 70). For large Eeynolds numbers^ where the influence of
Tiscosity is negligihle, one ohtains a distribution of the mixing length
l/r over y/r which is independent of the Ee-number (fig. 79). The
following interpolation formula can be given for this distrihution:
- = 0,11+ _ 0.08 f 1 - ^ ) - 0.06(1 - ^) (16.16)
In this equation y signifies the distance from the wall. The develop-
ment of equation (16.16) for small y/r (neighborhood of the wall) gives
2
I = o.kj - o.k-k — + . . . (16.16a)
In the neighhorhood of the wall the mixing length is, therefore, propor-
tional to the distance from the wall. Equation (16.I6) for the distri—
hution of the mixing length applies not only to the smooth pipe, hut,
according to the measurements of Nikuradse (reference 71) also to the
rough pipe, as can he seen from figure 79- From this fact one can derive
in a very simple manner a universal form for the law of velocity distri-
hution, valid for the smooth as well as for the rough pipe. One puts
for the mixing length distrihution: I = Kj f('^')with f ^^^-^ 1 for ^ -^' 0.
Furthermore follows from the linear distrihution of the shearing stress
over the cross section:
T = T^ (1 — ^j (y= distance from the wall)
together with equation (15-9)
<iy I Up
^ ^ (16.17)
" 7f(y/r)
and hence by integration:
MCA TM Wo. 1218 23
u = ^ J! 1_1 (16.18)
the lower limit of Intergration Jq where the velocity equals zero is,
according to the considerations of the previous section, proportional
to v/v^j thus: Jq/t = F-|_f— \ From equation (16.18) follows:
r \ r
and therefo'rej from equations (I6.I8) and (l6.l8a)
%ax - u = ^* F Cy/r)
(16.19)
This law, with the same fimctlon F(y/r), applies equally to smooth and
rough pipes. It states that the curves of the velocity distrihution over
the pipe cross section for all Eeynolds numbers and all roughnesses can
"be made congruent "by shifting along the velocity axis, if one plots
(u^q y — u/v^ against y/r (fig. 80). The explicit expression for the
function F(y/r) is ohtainsd immediately from equation (l6.1i<-)j according
to which ■
u - u = 2.5v. to ^ = 5.75 -^^ log - (16.20)
max jy
Universal Resistance Law
According to their derivation the velocity— distribution— law
(equations (16.I9) and (l6.20)) are to he regarded as valid for arbitrary
Eeynolds number since the laminar viscosity was neglected as compared
with the turbulent viscosity. We shall now derive from the velocity-
distribution— law equation (l6.20) a resistance law which, in contrast to
Blasius*, applies up to Reynolds numbers of arbitrary magnitude.
2k WACA TM No. 12l8
From equation (l6.20) one may determine "by integration over the
cross section the mean flow velocity u. One finds:
^ = "majc - 3.75 v» (l6.21)
The test results of Nikuradse (reference 70) gave a number slightly
different from 3-75^ namely:
According to equation (16.5):
X. = 8 (^) ^ (16.23)
From the uniyersal Telocity distribution law of the smooth pipe
equation (l6.l4) follows:
%ax = ^* {2-5 in ~ -^ 5.5
and hence the connection with equation (16. 2l):
u = T^ J2.5 In ^ + 1.75}* (16.24)
The Reynolds numher enters into the calculation by msans of the identity;
III ^ 1 ud T* ^ ud \[X
V - 2 V ^ V i^^
Thus results from equations (16.23) and (l6.2k):
8 •
X =
2.5 Zn (^^ \lxj - 2.5 In 1^ \|2 + 1.75
(2.035 log(^ \fx)-0.9l|
WACA TM No. 1218 25
or:
-^ = 2.035 log (^ ^) - 0.91 (16.25)
Accordingly a straight line must result for the resistance law of the smooth
pipe, if one plots l/ \/7, against logf^>/Xj. This is Tery well confirmed
hy Nikuradse's measurement (fig. 8I). The numerical values according to the
measurements differ only slightly from those of this theoretical derivation.
From Nikuradse's measurements was found:
(16.26)
Universal Resistance La v For Smooth Pipes
This is the final resistance law for smooth pipes. On the basis of
its derivation it may he extrapolated up to Reynolds numbers of arbitrary
magnitude. Thus measurements for larger Reynolds numbers than those of
Nikuradse's tests are not required. Up to Re = 100,000 this 'juiiversal
resistance law Is In good agreement with the Blasius law according to
equation (16.U). For higher Reynolds numbers the Blasius law deviates
considerably from the measurements (fig. 8I).
Concerning the determination of X from equation (16.26), where it
appears on both sides, it should be added that it can be easily obtained
by successive approximation.
Eleventh Lecture (February 16, 19i4-2)
b. The Rough Pipe
The ch.aracteristic parameter for the flow along a rough wall is the
ratio of grain size k of the roughness to the boundary layer thickness,
particularly to the thickness of the laminar sublayer 5 which is
always present within the turbulent friction layer in the immediate
neighborhood of the wall. The thickness of the laminar sublayer is
5, = number — . The effectiveness of roughness of a certain grain size
depends, therefore, on the dimensionlese roughness coefficient k/6,«,
In the experimental investigations of the resistance of turbulent flows
over rough walls_, the rough pipe has been studied very thoroughly since it
is of great practical importaiice. Besides depending on the Reynolds number,
the resistance of a rough pipe is a function of the relative roughness r/k.
One distinguishes for the resistance law of a rough pipe three regions.
26 NACA TM No. 12l8
The subsequently given "boundaries of these regions are Talid for sand
roughness kg like those investigated "by Nikuradse (reference 71).
1. Hydraulically smooth: The grain size of the roughness is so
small that all roughnesses lie within the laminar aulilayer. In this
case the roughness has no drag increasing effect. This case exists for
small Eeynolds numbers and for values of the characteristic roughness
numher: 2. — 5.
2. Fully developed roughness flow: The grain size of the roughness
is so large that all roughnesses project from the laminer auh— layer. The
friction drag then consists predominantly of the form drag of the single
roughness elements. A purely square drag law applies. For the pipe the
resistance coefficient X, is then independent of Re and only dependent
on the relative roughness k/r. This law exists for very large Resmolds
numbers. For sand roughness this law applies for: — — §■ ^ 70.
3. Intermediate region: Only a fraction of the roughness elements
project from the laminar suhlayer. The drag coefficient depends on r/k
as well as on Re. This law exists for medium Reynolds numters andj for
^ v^ ^
the sand roughness, for: 5 — 70.
The dependence of the pipe resistance coefficient on the Reynolds
number and on the relative roughneBS according to the measurements of
Nikuradse (reference 71) can he seen from figure 82 as well as, in
particular, the three laws Just given.
The velocity distrihution on a rough wall is given, hasically, in
the same way as for the smooth wall hy equation (I5.I6). One has only
to substitute for the constant of integration y^ another value: yQ
proportional to the roughness grain size. One puts for sand roughness
yQ = 7 kg and hence obtains from equation (I5.I6)
(16.27)
f
■V
u
1 1
= — ^
7,n
J
- 7,n
y ?
V*
K
^
^a
)
The constant 7 is, moreover, a function of the roughness form and the
roughness distribution. Comparison with experiments of Nikuradse
(reference 71) on pipes roughened artificially by sand yields for the
velocity distribution the general formula:
2.5 In ^ + B ] (16.28)
MCA TM No. 1218 27
the constant B teing different in each domain descrlhed above; it
depends on v^k /v .
For the fully developed roughness flow B = 8.5, thus:
u = v^
^2.5 to ^ + 8.5") (fully rough) (I6.29)
whence follows:
%ax = ^*(2-5 ^n^+ 8.5) (16.30)
and:
u - u = v^ 2.5 2n - (16.30a)
in agreement with equation (l6.20). Thus there applies also to rough
pipes, as equation (l6.21) did hefore to the smooth pipe, the relation:
'^ = %ax - 3-75 ^* (16.31)
From here one can, by a calculation which is perfectly analogous to the
previous one for the smooth pipe, easily arrive at the resistance law of
the rough pipe for fully developed roughness flow. By insertion of u^^^
according to equation (16.3O) into equation (I6.3I) one obtains:
or:
or;
u = v^ ("2.5 2n ^ + 1^.75) (16.32)
/v \2 8
x = Q[^) =, 2 (16.33)
(2.5 in ^.^.75)
X = -, Tp (16. 3i^)
(2.0 log — + 1.68j
28
NACA TM No. 12l8
This is the square resistance law of the fully deTeloped roughness flow.
Comparison with the test results of Nikuradse (fig. 83) shows that one
ohtains "better agreement if one changes the number 1.68 to 1.7i^. Thus
the resistance law of the pipe flow for fully developed roughness is:
(16.35)
X
1
(^
log
V
^s
+
1
-f
In the plots of -j= against log r/kg, (fig. 83) the test results fall
very aGcxirately on a straight line.
For flow along a rough wall in the intermediate region the constant
B in equation (16.28) is, moreover, a function of the roughness coeffi-
cient T^ /v . For this case also one can derive the resistancfe law
immediately from the velocity distribution. According to equation (16.28);
B = — - 2.5 2n -i =
u
^2£ _ 2.5 to ^
* 8
(16.36)
On the other hand, according to equations (16.3I) and (16.23)
u
max
^» ^*
- + 3.75 = ^+ 3.75
(16.37)
so that one ohtains from equation (I6.36)
B
V
u o I. -,_ y_ ^ 2V2 _ o c -,„ r
= ^^ - 2.5 In
V-. k
2.5 2n i^ + 3.75 (16.38)
^s
One can, therefore, determine the constant B as a function of v^k^S'
either from the velocity distribution or from the resistance law. The
plot in figure 8k- shows good agreement hetween the values determined 'dj
these two methods. At the same time the determination of the resistance
law from the velocity distrlhution is confirmed.
The formula for B includes the case of the smooth pipe. B is,
according to equation (l6.1i4-).
NACA TM Wo. 1218 29
u T "^^^s
B = — - 2.5 Zn f- = 2.5 In + 5.5 (16.39)
Thus a straight line results for B in the plot against log t^ /v.
Other Eoughnesses
Because of the great practical importance of the roughness— pr oh lem
a few data concerning roughnesses other than the special sand roughness
will he given. Fikuradse's sand roughness may also he characterized hy
the fact that the roughness density was at its maximum value^ because
the wall was covered with sand as densely as possible. For many practical
roughnesses the roughness density is considerably smaller. In such cases
the drag then depends, for one thing, on form and height of the roughness,
and, moreover, on the roughness density. It -is useful to classify any
arbitrary roughness in the scale of a standard roughness. Nikuradse's
sand roughness suggests itself as roughness reference (roughness scale)
because it was investigated for a very large range of Reynolds numbers
and relative roughnesses. Classification with respect to the roughness
scale is simplest for the region of fully developed roughness. According
to what was said previously, for this region the velocity distribution
is given by:
^ = 5.75 ZOg ^ + Bg Bg = 8.5 (16.40)
and the resistance coefficient by:
r \2
2.0 log — + 1.1k j
(16. Ul)
One now relates to an arbitrary roughness k an equivalent sand roughness
kg by the ratio
kg = a k (16.42)
Where by equivalent sand roughness kg is meant that grain size of sand
roughness which has, according to equation (l6.4l) the same resistance
as the given ro-oghness k.
Basically, of course, the equivalent sand roughness kg can be
determined by a reslstajice meas'jrement on the pipe. However, such measurements
30 NACA TM No. 12l8
for arbitrary roughneases are difficult to perform. Measijrements on
arl)itrary roughnesses In a tunnel with plane walls are more convenient.
To this purpose an exchangealile wall of a tunnel with rectangular cross
section Is provided with the roughness to he Investigated (fig. 85).
From the measurement of the velocity distrihution in sach a tunnel with
a rough and a smooth longitudinal wall one ohtalns, for the logarithmic
plot against the distance from the wall, a triangiolar velocity distrihu—
tion (compare fig. 85). From the logarithmic plot of the velocity dlstrl-
hutlon over the rough wall
u = n^ log y + m^ (I6.U3)
one ohtalns "by comparison with the universal law according to
equation (I6.28) for the shearing stress velocity at the wall:
*"" 5.75
Further, one determines for the roughness to "be investigated the constant
B of the velocity distrlhution law, namely:
B = — - 5.75 log I (I6.if5)
^#r ^
By comparison of equation (iS.k^) with (16.40) one ohtalns for the
equivalent sand roughness-:
]£
5.75 Zog -^ = 8.5 -B ' (16. 46)
k
In this way one may determine the drag for arhitrary roughnesses from a
simple measurement in the roughness tunnel. This jaethod may "be also
carried over to the case of the intermediate region.
CHAPTER XVTI. THE FRICTION DRAG OF THE FLAT PLATE
IN LONGrmOINAL FLOW
The tur"bulent friction drag of the plate in longitudinal flow is
of very great practical importance, for Instance as friction drag of wlngs^
airplane fuselages, or ships. The exact measurement of the friction drag
for the large Reynolds numhers of practice is extremely difficult. Thus
WACA TM No. 1218 31
it is particularly important that one can, according to Prandtl
(references 73 bxiA. 7^)? calculate the friction drag of surfaces from the
results of pipe flow studies. This conversion from the pipe to the plate
can he made for the smooth as well as for the rough plate.
a. The Smooth Plate
One assumes, for simplification, that the boundary layer on the
plate is turbulent from the leading edge. Let the coordinate system he
selected according to figure 86. The boundary layer thickness 6(x)
increases with the length of run x. Let b be the width of the plate.
For the transition from pipe to plate the free stream velocity Uq of
the plate corresponds to the maximum velocity u,^ in the pipe, and
the boundary layer thickness 6 to the pipe radius r.
One now makes the fundamental assumption that the same velocity
distribution exists in the boundary layer on the plate as in the pipe.
This is pertalnly not exactly correct since the velocity distribution in
the pipe is influenced by a pressure drop, whereas on the plate the
pressure gradient equals zero. However, slight differences In velocity
distribution are insignificant since it is the momentum Integral which
is of fundamental Importance for the drag. For the drag W(x) of one
side of the plate of length x, according to equations (lO.l) and (10.2)
¥(x) = b T (x) dx = bp
p5(x)
u (Uq - u) dy (17.1)
whence
1 dW
b dx
^o(x) (17.1a)
The equation (I7.I) can also be written in the form
il
W(x) = bpu/6(x)j -^ ^1 - -^j d i (17.2)
For the velocity distribution in the boundary layer one now ass^jmes
the 1/7-power law found for the pipe. Replacing u^^^ by U^ and r
by 5]^ one may write this law, according to equation (16.II):
32
NACA TM No. 12l8
= V
1/7
(17.3)
Hence the momentum. Integral "becomes
il 01
J =
u
1 -
t)
d i
s
.''%
l/7\
Tl« dV
^ (17A)
and thus
W(x) = ^ hpTj/5(x)
Hence follows^ according to equation (17.1a)
(17.5)
° - 72 P^o dl
(17.6)
On the other hand, one had found "before for the smooth pipe, equation (16.I3)
again replacing r "by 5 and u "by U :
TTWY " o
Tq = 0.0225 pUo'^^ (^^'
(17.7)
By equating equations (17.6) and (17.7) results:
7 2d5 "^A/vN^A
12 P^o i = °-°225 PTJ, (5
This is a differential equation for 5(x). The integration yields;
k 5A 72 /v\^A
_5 =-0.022,1-) X
>
(17.8)
/ X / v\l/5 i^/5
6(x) = 0.37(^1 ' X ^^
MCA TM No. 1218
33
5(x)
_ 0.37X
Ee,
V
(17.9)
For the turliulent "boundary layer the boundary layer thickness, therefore,
increases with x^/5. The corresponding equation for the laminar flow
was, according to equation (9.21a), 6 = 5\/vx/Uq.
By suhstitution of equation (17.8) into equation (17. 5) one ohtains
2 -^/5
W(x) = 0.036 hpUg X (Re^)
or for the drag coefficient c„ = W/^ U xb:
Cj. = 0.072 (Ee )
-1/5
Comparing this result with test results on plates one finds the numerical
value 0.072 to he somewhat too low.
c„ = 0.074 (Ee,^)
-1/5
valid for
5 X lo5 < Ee < 10'''
(17.10)
corresponds "better to the measurements. This law holds true only
7
for Ee < 10 , corresponding to the fact that the Blasius pipe resistance
law and the l/7— power law of the velocity distri'bution, which form the
hasis of this plate drag law, are not valid for large Eeynolds numbers.
This law is represented in figure 87 together with the laminar— flow law
according to equation (9.19)« The initial laminar flow on the front part
of the plate can he taken into consideration by a subtraction, according
to Prandtl (reference 73):
c-f = 0.07ij- Ee;j
-1/5
1700
Ee
5 X 105 < R
< 10
7
(17.11)
The plate drag law for very large Eeynolds numbers can be obtained
in essentially the same way by starting from the universal logarithmic
law for the velocity distribution equation (l6.l4-l) which, according to
3h
NACA TM No. 12X8
its derlTatioiij is valid up to Eeynolds numbers of arbitrary magnitude.
Here the calculation TDecoines considerably more complicated. The
development of the calculation is clarified if one first introduces the
Telocity distribution in a general form. We had introduced for the pipe
y^*
The values at
flow the dimensionless variables cp = -^ and ti
the edge of the boundary layer are to be denoted by the index 0, thus
^o =
5v.
^o =
(17.12)
Then:
^ = ^*^ = ^o ^
(17.13)
y =
dy
V^o
U
dT,
(17.14)
From the equation t = follows, with W according to equation (17.1)
o b dx
P ^* P dE <> ' ^(u. - ^) dy ;>
r 05
^^,
\^ =T^T-< ^(Uo - u) dy
dx dTio
''o
and according to equation (17.13)
U_
vcp
o d
9o
2 " ^o ^%
u(Uo - u)dTi
dx
NACA TM Wo. 1218
35
o _ Jo _d_
(p^2 " ^° dx dTlo
,no
t^-t'^^
^
" dX dTl^ ^O V "Pr
dn
U 6.1]
_Q_ = V —a -d—
cp^2 dx dri^
cp-^)d,
(17.15)
2\
In forming the integral -= — P f cp — ^ j drj one must note the following
facte: The differentiation with respect to the upper limit gives zero,
since for t\ = t)q, cp = cpQ. In the differentiation of the integrand 9
is to he regarded as constant, and cp as a function of t) . Therefore
•iTlo
rih
^1o
Tie
9
dT)
9
Thus follows from equation (17.15)
1%
dx dT),
(17.16)
One puts, for simplification.
dcp^
1%
F(T,^) = ^ I 9%
(17.17)
and obtains from equation (17.16)
36
NACA TM No. 12l8
^ = xf ^^^o) '^\
If one assijines this law to "be valid from ths leading edge of the plate
(x = 0), that iSj that the flow is turtulent starting from the front,
the integration gives:
X = X , (,^)
(17.18)
with
1^1
(m) = ^^^o)^
(17.19)
'ao=°
Equation (17. l8) can also he written so that the Ee-mxmber formed with
the length of run x appears:
U X
Ee^ = -^ = $ (r,^)
(17.20)
with:
(Til)
dcp
'%=°
R^.
9 dTi dTi
,"o
(17.19a)
Equation (17.20) gives the relation hetween ths dimensionless houndary
layer thickness t^ = v^b/v and the Re-number U x/v .
The drag remains to "be calculated. From
follows J "because ^ - \> P r ^ dji
NACA TM No. 12l8
37
1^1
¥ = -b p Uo V
F(Tlo)
Jt1q=o
^r
dTiQ = b p Uq vt (ti^)
(17.21)
where
t
(tI^)
P^l
F(no)
^,
dTi
2 'o
n.=°
(17.22)
/p 2
The drag coefficient c = ¥/^ Uq "b x "becomes finally
c. = l^i, . = 2
^ V (Til) «(t1i)
(17.23)
Hence C|. also turns out to he a function of t]-,. Equations (17.20)
and (17.23) together give a parametric representation of c as a function
^ v„5
of Ee-jj-j where the parameter is the boundary layer thickness r\ = — ^— .
Numerical Results
In order to arrive at numerical results, one must introduce a
special function for 9(1)). For the l/7— power law according to equation
(16.11) that is, with cp = 01]/', one would obtain the drag law according
to equation (17.IO). One uses the universal logarithmic velocity distri-
bution law, equation (l6.l4).
cp = 2.5 In T] + 5.5
In order to make the carrying out of the integrations more convenient,
one writes
cp = 2.5 In (1 + 9ti)
Then 9 becomes, for t) = 0, cp = 0. The adding of the one changes
cp(ti) a little, only for very small r\ and has only litble influence
on the integrals. If one writes the law at first in the general form
38
• NACA TM No. 12l8
cp = a 2n (1 + "bT))
(17.21+)
the calculation of the Integrals equations (17.17)^ (17. 19)^ (17.22),
with z = 1 + Tdt] yields
F(t]) = a^ (zn^z _2toz + 2--
^(t) = — (2 2n z — kz Zn z — 2 In z + 6z — 6 j
^^) = =h S^ + 1 -
2(z -1)
2n z
> (17.25)
With the numerical values
a =2.493 t = 8.93
one ohtains for the drag lav the following tahle:
103
-6
Ee 10
c 10^
f
0.500
0.337
5.65
1.00
0.820
J+.75
2.00
1.96
i+.05
3.00
3.25
3.71
5.00
6.10
3.3i^
12.0
17.7
2.81
20.0
32.5
2.57
50.0
96.5
2.20
100.0
217.5
1.96
500.0
li+Ol.O
1.55
This tatle can he replaced by the following interpolation formula:
Cj =
0.472
(Zog Ee^.)
2.58
NACA TM No. 12l8
39
Comparison -with test resiilts shows that the agreement improTes if the
number 0.472 is slightly varied, hy putting
°f =
O.k^^
(log Ee^)
2.58
6 9
10 < Ee^< 10^
(17.26)
Prandtl - Schlichtlng's Plate Drag Law
The laminar approach length may again "be taken into consideration
"by the same sobtraction as before; thus:
Cf =
OA55
_ ITOO
(log Ee^) ^ ^
5 X lo5
< Ee^ < 10--
(17.27)
Whereas the system of formulas equation (17.25) Is valid up to Ee-^iumbers
of arbitrary magnitude, the interpolation formulas, eq^uations (I7.26)
and (17.27), have the upper limit Ee = 10 . However, this limit taJces
care of all Ee— numbers occurring in practice. The theoretical formula
(equation (17.27)) is also plotted in figure 87. Figure 88 gives a
comparison with test results on plates, wings, and airship bodies. The
agreement is quite good.
Very recently this plate drag law has been somewhat improved by
Schultz-Grunow (reference 89). Until then, the turbulent velocity profile
measured in the pipe (l/7— power law, logarithmic law) had been carried
over directly to the plate, mainly because accurate velocity distribution
measurements of the plate boundary layer did not exist. The exact
measurement of the plate boundary layer showed, however, that the plate
profile does not completely coincide with the pipe profile. The teat
points show, for large distance from the wall, a slight upward deviation
from the logarithmic law found for the pipe. Thus the loss of momentum
on the plate is somewhat smaller than that calculated with the logarithmic
law. Schultz-G-runow repeated the calculation of the drag law according
to the formula system given above with the velocity distribution law
for the plate measured by him. His result Is represented by the
interpolation formula
°f =
O.lj-27
(- 0.1+07 + ^og Ee„)
2.61+
6
10 < Ee,
< 10'
(17.28)
I^Q NACA TM No. 12l8
This law Is also plotted in figure 87. The differences from the Prandtl-
Schlichting law are only slight.*
The corresponding rotationally— symmetrical problem, that is, the
turbulent boundary layer on a body of revolution at zero incidence, was
treated by C. B. Millikan (reference 79). The l/7-power law of the
Telocity distribution was taken as basis. Application to the general
case has not yet been made.
Twelfth Lecture (February 23, 1914-2)
b. The Rough Plate
The conTersion from pipe resistance to the plate drag may be carried
out for the rough plate in the same manner as described previously for the
smooth plate. One assumes a plate uniformly covered with the same rough-
ness k. Since the boundary layer thickness 5 increases from the front
toward the rear, the ratio k/5 which is significant for the drag decreases
from the, front toward the rear. Behind the initial laminar run, therefore,
follows at first the region of the fully developed roughness flow; the so-
called Intermediate region follows and farthest toward the rear there is,
finally, if the plate is long enough, the region of the hydraullcally
smooth flow. These regions are determined by specification of the n-umerical
values for the roughness coefficient v^k /v. In order to obtain the drag
of the rough plate, one must perform the conversion from pipe flow to plate
flow for each of these three regions individually. This calculation was
carried out by Prandtl and Schlichting (reference 76), based on the results
of Nikuradse (reference 71) for the pipe tests with sand roughness. For
this conversion one starts from the universal velocity distribution law of
the rough pipe according to equation (16.28), the quantity B being
dependent also on the characteristic roughness value v^k /v^ according
to figure Qk. The calculation takes basically the same course as described
in detail for the smooth plate in chapter XVTIa. It is, however, rather
complicated and will not be reproduced here. One obtains as final result
for the total drag coefficient of the sand— rough plate a diagram (fig. 89)
• which represents the drag coefficient as a function of the Reynolds
number U^Z/v with the relative roughness Z/kg as parameter. Just as
for the pipe, a given relative roughness 2 /kg has a drag Increasing
effect not for all Ee-numbers, but only above a certain Re-Hiumber. This
diagram is applicable also for roughnesses other than sand roughness, if
one uses the equivalent sand roughness. In the diagram (fig. 89) the
square drag law is attained. Just as for the pipe, for every relative
* The tables pertaining to the plate drag formulas are given in
table 7 5 chapter XXII.
NACA TM No. 12l8
kl
roughness 2 As provided the Ee-number is sufficiently large,
interpolation formula
The
Cj = [1.89 + 1.62 Zog ^ j
2 A-2.5
(17.29)
applies to this law.
c. The Admlssitle Eoughness
The problem of the admissible roiighness of a wall in a flow is
very important in practice since it concerns the effort that might
reasonably be expended in smoothing a surface for the purpose of drag
reduction. Admissible roughness signifies that roughness above which a
drag increase would occur in the given turbulent friction layer (which,
therefore, still is in effect hydraulically smooth). The admissible
relative roughness kg/z decreases with increasing Re— number UqZ/v
as one can see from figure 89. It is the point where the particular
curve Z/kg diverges from the curve of the smooth wall. One finds the
values for the admissible relative roughness according to the following
table; they can also be combined into the one formula
^o ■'^s admlss.
102
(17.30)
U I
V
lo5
xo«
10^
108
105
admlss.
10-3
10-^
10-5
10^
10-^
From equation (17.3O) one recognizes that the admissible roughness height
is by no means a function of the plate length. This fact is significant
for instance for the admissible roughness of a wing. Equation (17.3O)
states that for equal velocity the admiesible roughness height is the
same for a full scale wing as for a model wing. Let us assume a numerical
example :
Wing: chord 2 = 2m
Velocity U = 3OO km/h = 83 m/sec
k2 KACA TM No. 12l8
From equation (17. 30) results an admissible magnitude of
roughness kg = 0.02 mm. This degree of smoothness is not always attained
"bj the wing surfaces manufactured in practice, so that the latter have a
certain roughness drag. In the considerations just made one deals with
an increase of the friction drag in an a priori turbulent friction layer.
However, the roughness may also change the drag "by disturbing the
laminar friction layer to such an extent that the point of laminar/
turbulent transition is shifted toward the front. Thereby the drag can
be increased or reduced according to the shape of the body. The drag is
increased by this displacement of the transition point" if the body in
question has predominant friction drag (for instance wing profile). The
drag might be reduced, circumstances permitting, for a body with pre-
dominant pressure drag (for instance, the circular cylinder). One calls
the roughness height which causes the transition the "critical roughness
height". According to Japanese measurements (reference 77) this critical
roughness height for the laminar friction layer is given by
^*^crlt
* -^^^ = 15 (17.31)
A numerical example follows:
Assume, as prescribed before, a
wing Z = 2m
U = 300 km/h = 83 m/sec
then Re = UqI/v = lo'^. Consider the point of the wing x = 0.12, thus
Re^j. = U x/V = 10 . Up to this point the boundary layer might remain
laminar under the effect of the pressure drop. The wall shearing stress
for the laminar boundary layer is according to equation (9.17)
Tq ^ ^^^ ^^ 2.rir" 6900 m^
-^ = 0.332 Uq \^-^ = 0.332 -^— = 2.29 p
P VUo^ 103 sec^
hence:
T* = V^o/P = 1*52 m/sec
and according to equation (17.3I)
k = 15 T^ = -^^- i 10'~'^ = 0,14 mm
crit ^* 1.52 7
NACA TM No. 12l8 k3
The critical roughness height causing the transition Is^ therefore^ ahout
ten times as high as the roughness height admissible In the turhulent
friction layer. The laminar friction layer therefore "tolerates" a
greater roughness than the turhulent one.
The following can he said ahout the Influence of the roughness on
^the form drag: Sharp— edged "bodies are Indifferent to surface roughnesses
"because for them the transition point Is fixed hy the edges, as for
Instance for the plate normal to the flow. Short ciirved bodies, on the
other hand, as for instance the circular cylinder, are sensitive. For
the circular cylinder the critical Reynolds number, for which the known
large pressure drag reduction occurs, is largely dependent on the
roughness. With increasing relative roughness k/R (E = radius of the
circular cylinder) Ee^j^j^^ decreases. According to British measurements
(reference 9O) "tiie drag curves for a circular cylinder with different
relative roughnesses have a course as indicated in figure 90. The
boundary layer is so disturbed by the rot^hness that the laminar /turbulent
transition occurs for a considerably smaller Ee-number than for the
smooth cylinder. The roughness has here the same effect as Prandtl's
trip wire, that is, in a certain region of Ee-^iumbers it decreases the
drag. It is true, however, that the supercritical drag coefficient is
then always larger for the rough circular cylinder than for the smooth
one.
CHAPTER XVTII. THE TUEBULENT ERICTICJN LAYER IN
ACCELEEATED AND RETARDED FLOW
The cases of turbulent friction layer treated so far are relatively
simple insofar as the velocity outside of the friction layer along the
wall is constant. Here as for the laminar flow the case of special
interest is where the velocity of the potential flow is variable along
the wall (pressure drop and pressure rise). As for laminar flow, the
form of the boundary layer profile along the wall is variable. In
practice this case exists for Instance for the friction layer on the
wing, on the turbine blade, and in the dlffuser. Of special interest is
the question of whether separation of the boundary layer occurs and,
if so, where the separation point is located. The problem consists
therefore for a prescribed potential flow in following the turbulent
friction layer by calculation. The calculation of the turbulent friction
drag is of Importance. The corresponding problem for the laminar friction
layer was solved by the Pohlhausen method (chapter X).
For the turbulent friction layer the method of Gruschwitz
(reference 78) proved best. Gruschwitz makes the asumption that the
velocity profiles of the turbulent boundary layer for pressure drop and
hk NACA TM No. 12l8
rise can be represented as a one— parameter family, if one plots u/u
against y/-a. -d signifies the momentum thickness which is, according
to equation (6.32), defined "by:
U^^ =1 u(U - u) dy (18.1)
As form parameter one selects
(18.2)
u('3) denoting the velocity in the friction layer at the distance from
the wall J = -a. That t[ actually is a serviceable form parameter can
he recognized from figure 9I where a family of turbulent boundary layer
profiles is plotted according to Gruschwitz. Gruschwitz found from his
measurements that the turbulent separation point Is given by
Ti = 0.8 (Separation) (I8.3)
The form parameter r] is analogous to the Pohlhausen— parameter X of
the laminar friction layer. However, a considerable difference exists
between -q and X: for the laminar friction layer an analytical relation
exists between X and the press''jre gradient and the boundary layer
thicknessj namely according to equation (10.41)
g2
X = -^^ (18. i^)
V dx
Such a relation is thus far lacking for the t^lrbulent boundary layer,
since one does not yet possess an analytical expression for the turbulent
velocity profiles*. One needs therefore an empirical equivalent for
equation (l8.J|).
For the special case of the turbulent friction layer without
1/7
pressure gradient where the l/7— power law u/U = (y/5) applies for
the velocity profile, one finds from equations (I8.I) and (l8.2)
* Compare, however, chapter XXIIb, where under certain assijmptions
such an analytical connection is indicated.
NACA TM Wo. 1218
i^5
1
6
J_
72
^ = -L T) = 0.1+87
(18.5)
Since in the case of the turhulent "boundary layer, an analytical
expression for the velocity dlBtrnDutlon does not exist, the calculation
is limited to the determination of the four characteristics of the friction
layer: form parameter t], wall shearing stress t displacement
thickness 5*, momentum thickness t3 . Four equations are required for
their calculation.
As for the laminar 'boundary layer, the momentum theorem yields
the first equation; the momentum theorem may, according to equation (10. 36)
be written in the form:
(I)
(18.6)
The second equation is yielded "by the function
T = ^^^)
(II)
(18.7)
obtained by Gruschwitz by evaluation of the measured velocity profiles
(fig. 92) J and regarded as generally valid. It can be derived also by
calculation from the form of the velocity profile (compare appendix
chapter XXIl) and yields:
Tl = 1 -
H - 1
H(H + 1)
H-1
(18.8)
The third equation is empirically derived by Gruschwitz from his
measurements. He considers that the energy variation of a particle moving
parallel to the wall at the distance y = i3 is a function of u^, U, t3, V.
Dimension considerations suggest the following relation:
^^=F(„Re)
q dx
(18.9)
= |n2
y = ^
and g-, = p + § u^^ signify the total pressure in the layer
The evaluation of the test results showed that a dependence on
1^6
NACA TM Wo. 1218
the Ee— number is practically non-existent, and that one can represent
equation (18.9) in the following manner:
■^ i = O.OO89U T] - O.OOlj-61
<1 djc
(18.10)
Furthermore, the identity
go -
V^^^-V
P 2
2^^
= £u2fl-]£i i.ciT,
if
is valid. One puts
1 T] = ^
■ (18.11)
and has therefore
dg-L
dx
dC
dx
(18.12)
Now equation (I8.IO) can "be written:
d^
■a ^ = - 0.00894 ^ + 0.00i|6l q
(III) (18.13)
The fourth equation is still missing and is replaced by the following
estimation of Tq: According to the calculations for the plate in
longitudinal flow, equation (IT-T) was:
^ - 0.0225(— j = 0.0225 (Eeg)'
-lA
(18.14)
If one takes into consideration that for the l/7— power law of the Telocity
distribution:
NACA TM No. 12l8
^7
6.=|5,
i3 = -1- 5
72
one can write equation (l8.lU) also;
Tn -lA -lA
° - 0.01338 (Reg^) = 0.01256 (Ee^)
01/
(17) (18.15)
For calculation of -a and t] (t, = Q. r\, respectlTsly) one must now solve
the following system of equations:
— + 0.00894 -^ = 0.01+61 i
dx V V
i^ ^ A ^ H\ .fl dg
djc I 2y q djc
PU^
(18.16)
2 T
= •^17 is a given function of x; H and — ^
2 pU^
are given functions
of T] = ^/q or i3, respectively. This system of equations is to be
solved downstream from the transition point.
Initial values: As initial value for i3 one takes the value from
the laminar friction layer at the transition point:
i3
oturh.
•a
olam.
(18.17)
This is "based on the consideration that the loss of momentum does not
vary at the transition point since it gives the drag. The initial value
of r\ is somewhat arbitrary. G-ruschwitz takes
T1_ = 0.1
and states that a different choice has little Influence on the result.
With these initial values the system of equations (18.16) may he
solved graphically, according to a method of Czuher (compare appendix.
Chapter XXII, where an example is given) . A first approximation for i3
is ohtained by first solving the second equation with constant values
for Tq /pU and H;
1^8 WACA TM No. 1218
L2.
2
PU
= 0.002; H = 1.5 (18.18)
are appropriate. Thsre'by the second equation Is a differential eq^uation
of the first order for -d. This first approximation i3t (x) is then
suhstituted into the first equation, vhich then 1)60011168 a differential
equation of the first order for ^(x); let its solution te denoted
hy L(x). Thus one has also a first approximation for tj: t]-,(x).
With T|, (x) one determines the course of H(r|) according to figure 92
and is now atle to improve t according to equation (I8.I5). These
values of hoth H and t are now inserted in the second equation, and
a second approximation -Boi^) is obtained. By sutstitution of -SoCx)
into the first equation one ohtains the second approximation ^p(x), etc.
The method converges so well that the answer is essentially attained in
the second approximation.
The separation point is given ty
Ti = .0.8
Incidental to the houndary layer calculation one obtains the following
characteristic values of the friction layer as functions of the arc
length X :
■bM, b*(x), Ti(x), To(x).
The "boundary layer calculation for the profile J OI5, c^^ = is given
as example in figure 93 • The transition point was assumed at the velocity
maximum. The calculation of the laminar "boundary layer for the same case
was Indicated in chapter XII. The details of this example are compiled
in the appendix, chapter XXII.
It should "be mentioned that the calculation for the tur'bulent
"boundary layer must "be performed anew for every Ee-numier UqI/v, whereas
only one calculation was necessary for the laminar "boundary layer. The
reasons are, first, that the transition point travels with the Ee-number,
and second, that the initial val ue of i3/t varies with Ee, since for
the laminar boundary layer t- \ — ^=^ at the transition point Is fixed.
"t y V
It must be noted that the values obtained for Tq become incorrect
in the neighborhood of the separation point: At the separation point t
must equal zero, whereas equation (18.I5) gives everywhere t / 0.
MCA TM No. 1218 kg
B oundary Layer Without Pressure Gradient
In this case q(x) = Constant. Equation (18.I3) can te written:
1 i^ = _
- ^ = - o.oo89ij-Ti + 0.001^61 (18.19)
or, "because q.(x) = constant;
^ = iMl = q U (18.20)
djc dx ^ djc
Thus equation (18.I9) becomes;
^ ^ = - 0.0089UT1 + o.ooi)-6i (18.21)
A solution of this equation is:
. 0,00461 ^ g (^8.22)
' 0.0089!^ ^ ^
Since at the ■beginning of the tiirbulent friction layer t\ is smaller
than this value (transition point t) = 0.1) and since according to
equation (l8.2l) drj/djc > 0, t] must in this case approach the value
T] = 0.516 asymptotically from "below. For the velocity profile of the
1/7— power law, t\ = O.U87 (compare equation (18.15)). The profile
attained asymptotically for uniform pressure (p = constant) therefore
almost agrees with the l/7— power law that was previously applied to the
plate in longitudinal flow.
A great many "boundary layer calculations according to this method
are performed in the dissertation "by Pretsch (reference 80).
Thirteenth Lecture (March 2, 19^12)
CHAPTER XIX. FREE TURBULENCE
a. General iRemarks; Estimations
After considering so far almost exclusively the turbulent flow
along solid walls, we shall now treat a few cases of the so— called free
turbulence. By that one understands turbulent flows where no solid walls
50 WACA TM No. 12l8
are present. Examples are the spreading of a Jet and its mixing with the
surrounding fluid at rest; or the wake flow behind a "body towed thi^ough
the fluid at rest (fig. 9^)- Qualitatively these turbulent flows take a
course similar to that for the laminar case (compare chapter IZ) ; quanti—
tatively, howerer, considerable differences exist, since the turbialent
friction is very much larger than the laminar friction. In a certain way,
the cases of free turbulence are, with respect to calculations, simpler
than turbulent flows along a wall, since the laminar sublayer is not
present and the laminar friction as compared with the turbulent one can
therefore be neglected for the entire flow domain. The free turbulence
may be treated satisfactorily with Prandtl's concept of the turbulent
shearing stress according to equation (I5.9):
.1^
du
Su ,
— (19.1)
oy
the mixing length I being assumed a pure position function. The
turbulent friction has the effect of making the Jet width increase and
the velocity at its center decrease with increasing distance along the
Jet.
We now perform rough calculations, according to Prandtl (reference 2,
Part I), for a few cases of free turbulence which give Information about
the laws governing the increase of width and the decrease of "depth" with
the distance x.
It has proved useful for such turbulent Jet problems to set the
mixing length I proportional to the Jet width b:
1 = p = constant (19.2)
b
Furthermore, the following rule has held true: The increase of the
width b of the mixing zone with time is proportional to the fluctuation
of the transverse velocity v' :
^~v. (19.3)
, D S S
D/Dt signifies the substantial derivative: thus: — = u ^r— + v :r— .
Dt dx dy
According to our previous estimation, equation (I5.5): v' = 2
Therefore :
MCA TM No. 1218 51
Dt dy
Furthermore, the mean value of ^ equals approximately:
- number — =^
Sy
and thus:
=rr = nuuiber t- u = number B u (I9.5)
Dt D max max v ^ v;
Jet (Plane and Circular]
We shall estimate , "by means of these relations, how the width
increases with the distance x and the velocity at the center decreaeea.
At first, for the circular as well as for the plane Jet:
=rr = number u -r- (19 -o)
Dt max dx \ ^ j
It follows, "by comparison with equation (I9.5):
-T— = number r- = number
dx b
b = number x + constant
If the origin of coordinates is suitably selected (it need not coincide
with the orifice) one has therefore:
b = number x
(plane and circular Jet) (I9.7)
The relation between u^^y and x Is obtained from the momentum
theorem. Since the pressijire is constant, in the x— direction, the x—
momentum must be independent of x, thus:
J-p/u^dF = constant
52
NACA TM Nc 12l8
whence follows for the circular jet:
or;
2 2
J = nuniber p u b
u = niniber t- \| —
max ^ \ P
and "because of equation (I9.7)
^^max
= number
X Yp
(circular Jet) (19.8)
For the plane Jet, if J' signifies the momentum, per unit length of the
Jet:
J' '= numher p u h
1 y J'
"^^ = ^™*er — \) -p-
y^
and because of eq.uation (I9.7):
(plane Jet) (19-9)
Wake (plane and circular)
The calculation for the wake is somewhat different, since the
momentum which gives directly the drag of the "body must he calculated in
a slightly different way. The momentum integral is now (compare
equation (9.14.0)) :
W=J=p/u(Uo-u)dF
(19.10)
At large distance from the body u* = Uo — u is small compared with U(-
(fig. 95) so that u» « Uq and
NACA TM No. 12l8
53
u(Uo - u) = (Uq - u' ) .u» Z Uq u«
Thus one obtains for the circiilar wake:
§ c,., F U^2 «, p TJ u« jt h2
2 w ■" "'o
t c F
u* w
— — /w — ^^—
o no
(19.11)
Instead of equation (19.6) now applies:
Dt ° dbc
(19.12)
and instead of equation (19.5)
=rr = number t- u'
Dt D
(19.13)
Equating of equations (19.12) and (19.I3) gives:
tJ ^ ~ i u' = p u»
o djc l3
(19.1^)
By comparison with equation (I9.II) one obtains:
dx jt
p c^ F X
(Wake Circular)
(19.15)
By insertion in equation (I9.II) results:
u' .. 1 /"v ^
^o 4p2,2
1/3
(Wake circular )
(19.16)
'^h
liACA TM No. 1218
For the plane wate "behind a long rod, wing, or the like with the
diameter d and the length L,W = c — U^, Ld and
¥ = J^pUqU'13L
and hence;
c d
(19.17)
and from this in combination with equation (I9.IU)
2 h — ~ p c d
dx w
h ~ \fpTT7
(wake plane)
(19.18)
By substitution in equation (I9.I7) results;
^c
d^
^1/2
u'
1
w
/V
—
—
Uo
2
^p
^/
/
(Wake plane)
(19.19)
ThuSj for the circular wake, the width increases with \^y^, and.
-2/3
the velocity decreases with x ; for the plane wake, the width
increases with \f^, and the velocity decreases with x ' .
The power laws for the width and the velocity at the center are
compiled once more in the following tahle. The corresponding laminar
cases which were partially treated in chapter IX, are included. More-
over, the case of the free Jet boundary is given, that is the mixing of
a homogeneous air flow with the adjoining air at rest. (Compare
figure 97. )
NACA TM Wo. 1218
55
Lami nar
TurlDulent
Case
Width
Velocity at Center
u
max
or u'
respectively
Width
h
Velocity at Center
u
max
or u'
respectively
Plane Jet
.2/3
X-V3
X
x-1/2
Circular Jet
X
x-1
X
x-1
Plane wake
xl/2
x-1/2
xl/2
x-1/2
Circular wake
xl/2
^1
xl/3
x-2/3
Free Jet "boundary
xl/2
x°
X
x°
POWER LAWS FOE THE INCEEASE WUH WIDTH AND THE DECREASE OF TELOCITY
WITH THE DISTANCE x
For a few of the cases treated here the velocity distribution will
he calctilated explicitly "below. The calculation on the "basis of the
Prandtl mixing length theorem was performed for the free Jet "boundary,
the plane Jet, and the circular Jet "by W. Tollmien (reference 81), for
the plane wake "by H. Schlichting (reference 82) and for the circular
wake "by L. M. Swain (reference 83).
The equations of motion for the plane stationary case are, according
to equation (lU.lU), if the laminar friction terms are completely
neglected:
= _i^ + i^+i ^Ih.
^ p ^ p Sx p Sy
u ^ + V ^^
Sv Sv
Sx hj
Su Sv
S^ 3y
1 M+ 1 ^+ 1 ^
p ^ p dx p Sy
(19.20)
y
56 MAC A TM No. 12l8
Td. The Plane Wake Flow
\Je shall now calculate the velocity diatrihution for the plane wake
flow. A cylindrical "body of diameter d and bt.zii h is considered.
Further, let
u' = U^ - u (19.21)
he the wake velocity. One applies the momentum theorem to a control
area according to figure 95 ^ "the rear ho-jndary B C of which lies at
such a large distance from the hody that the static pressure there has
the undisturhed value. As shown in detail in chapter H., equation (g.kO),
one ohtains:
W = h p / u (Uq - u) dy
BC
(19.22)
¥ = h p / (Uq - u' ) u' dy
J
BC
For large distances "behind the "body u' « Uq so that one may
2 , .
approximately neglect the term u» in comparison with 17^ u' m
equation (19.22). Hence:
(19.22a)
Since, on the other hand W = c„ d h ^ U "", there "becomes:
•' " 2 o
i+h
^' iy = I c^ i u^ (19.23)
-b
This prohlem can "be treated "rfith the boundary layer differential equations;
they read according to equation (I9.2O) with the Prandtl expression for the
turhulent shearing stress according to equation (I5.9), with p and a^
neglected:
SiACA TM Wo. 1218
57
^
Su Su 1 St ^,2 Su S a
u — + V — = = 22 TT
Sx By ^ Sy
5u I 5v _ Q
Sy By'
Sx Sy
/
For ths mixing length one puts, according to equation (19.2)
(I9.2J+)
2 = p b
(19.25)
Further :
U' = U-, + Up + .
^ = h
(19-26)
(19.27)
For the wake "velocity u-i and the width "b the power laws for the
decrease and increase, respectively, with x were already foimd in
equations (19.I8) and (19,19), One therefore writes:
u
-1/2
u ych^ ^^^)
o \ w
1
(19.28)
h = B (cwd x)
1/2
(19.20)
According to equations (19.27) and (19.29)
Sn
_ 1 Bti
^y B(cwd x)^/2 ^^
It
2 X
The estimation of the terms in equation (I9.2ij-) with respect to their
order of magnitude in x gives:
Bu -3/2 bv -3/2 Su -1 S u _ -3/2
^:— ~x'; — ~x-^'; — ~x ; ~ x -^' ;
Sx By ' Sy .2
V ~ X
*The terms u^ • - • signify additional terms of higher approximation,
which disappear- according to a higher power of x than does u-^.
58
WACA TM No. 1218
Hence the term v ^ ^ x whereas the largest terms ~ x .
equation (l9.2i|-) is simplified to:
1 2 ^^1 ^ ^1
- U — - -^2 1
° Sx
^ ^'
Thus
(19.30)
The neglected terms are taken into account only in the next approximation.
The further calculation gives:
^^1 / X \-l/2
S;
c d
_ i a f I - 1. f
2 X 2x
SU'
Sy
= u.
^-1/2
\\i J
f
B(c^dx)
1/2
S^u
^y
1 n r-^V'^'f" ^
w
° ' c d ) ^2
B c d X
w
2 Z
2 Bu-, S U-,
2 p^ U^^/^-^^^ ^ f'f"
!/c c
B/'c^dxj
1/2
After Insertion in equation (I9.3O) and eliminating the factor
2 1 / X \"^/^ / N
U — (-rV) "the following differential equation results for f(Ti)
i (f + Ti f) = ^ f«f"
B
(19.31)
The houndary conditions are:
Su
y = h: u = 0; — -
■^ dy
NACA TM No. 12l8 59
That is:
Ti = 1: f = f • = (19.32)
Th3 differential— eq^uatl on (I9.3O) may immediately te integrated once
and gives:
2
ifTi= — f'^ + Constant
2 B
Because of the hoondary conditions, the integration constant must equal
zero; thus:
1 3 2
- f Ti = — f '
2 ^ B .
This may he Integrated in closed form:
^1/2 1/2 ^ \ |2pf df
B dTi
df J B ,^ ,
if V2P'
2fl/2 = \ pi 2 3/2 ^ c„^g^3^^
-^N5^"^^^'
Because f = for ti = 1, C = - i J^ and hence:
3 y2p2
,2
= l^(l_,^3/2) ■ (19.33)
q _„2 \ /
f = i
9 2p2
60 NACA TM Wo. 12l8
The condition f = for T] = 1 is simultaneously satisfied according
to equation (19'33)« In. f", that Is — —, a singularity results at the
Sy2
center (t) = O) and on the edge. For ti = 0, f" =oo; the velocity
profile there has zero radius of curvature. At the edge there exists
a discontinuity in curvature. In contrast to the laminar "boundary
layer solutions, where the velocity asymptotically approaches the value
of the potential flow, one obtains here velocity profiles which adjoin
the potential flow at a finite distance from the center.
The constant B remains to he determined:
n+h Oh ph ni
u' dy =*2 u' dy = 2 / u^ dy = 2U^ c^d B / f(Ti) dii
J-h Jo t/o Jo
From equation (19-33) one finds: 2 / (^ — 'H ) '^'H = iT^
UO
and hence;
and, by comparison with equation (19-23)
or:
b2 1
= i; B = \/lO p (19.3^)
2 2
Thus the final result for the width of the wake and the velocity
distribution from equations (19-28) and (19.29) is:
'^Eeviewer's note: Integrating from —1 to +1, as was done in the
original German version, results in an imaginary term, which was avoided
in the translation by integrating from to +1 and doubling the result.
WACA TM No. 1218 61
"b = \ 10 (3 /"c d zS^'^
^1 .. VlO / X \-l/2
2 (19.35)
Uo 18P \c^^
1-(J''^'
The constant 3 = z/b is the only empirical constant of this theory; it
must "be determined from the measurements.
Comparison with the tests of Schlichting (reference 82) shows that
the two power la^-s (equations (I9.28) and (I9.29)) are well satisfied,
and also bhat the form of the velocity distrihution shows good agreement
with equation (19.35)^ figia^e 96. The constant 3 is determined as
p = 1 =,0.207
The solution found is a first approximation for large distancesj according
to the meas'jrements it is valid for x/c^d > 50' For smaller distances
-1 -1 /2
one may calculate additional -terms which are proportional to x , y: ' ,
. . . for the wake velocity in equation (19.26).
The rotationally— symmetrical wake problem was treated by
Miss L. M. Swain (reference 83). For the first approximation results
exactly the same function for the velocity distribution; only the power
laws for the width b and the velocity at the center u^' are different,
namely b «' x ' and u^' ~ x ' , as already Indicated in
equations (19.15) aJ^i (I9.16).
Fourteenth Lecture (March 9, 19^2)
c. The Free Jet Boundary
The plane problem of the mixing of a homogeneous air streaiu with
the adjoining air at rest shall also be treated somewhat more accurately
(fig. 97). It is approximately present for instance at the edge of the
free Jet of a wind tuiinel. The problem was solved by Tollmlen
(reference 8I).
The velocity profiles at various distances x are affine. One
Gets
u = U^f(Ti) = UJ'(ti) (10.^,6)
62 NACA TM No. 12l8
where
Ti = |; (t~ x) (19.37)
and
nr\) =/f(n) ^T
Furthermore set
2 = 01 = 01) (19.38)
The equation of motion reads:
u^+T^ = i^ = 2l2^^ (19.39)
Sx dy p Sy Sy ^ 2
One integrates the continuity equation hy the stream function:
ilf=j'udy=U^xj' f(Ti) dTi = U^ X F(ti) (I9.i^0)
Then:
|li = __£^F"; ^ = ^Y"; ^ = ^F'"
ax X ' ' By X ' ^^2 ^2
T = -^ = -U^ (F - TiF') (19.J^0a)
Suhatitution into the equation of motion (19.39) gi"^es, after division
hy Uo^/x;
FF" + 2c^ F"F'" = (19.^1)
NACA TM No. 12l8
63
The boundary conditions are:
at the inner edge:
Tl = \'
u = U :
F» = 1
1^-
F" =
T = 0:
F = Tl
at the outer edge:
T) = TI2:
u = 0:
F« =
^^ = 6:
oy
F" =
> (19-^2)
J
Since the "boundary points r\^ and T)p are still free, these five hoimdary
conditions can Tae satisfied by the differential equation of the third order
equation (I9.J4-I). By introduction of the new variable:
Tl* =
^y^
(19A3)
the differential equation (19.41) is transformed into (' = differentiation
with respect to t)*)
F F" + F" F'" =
(19.i^i^)
The solution F" = 0, which gives u = Constant, is eliminated. The
general solution of the linear differential equation
F + F'" =
(l9.i+5)
is
F = e
\-x\>
with \ signifying the roots of the equation X,-^ + 1 = 0, thus:
6U NACA TM Wo. 12l8
Hence the general solution is:
F =
r^
^
VJ„J , . 2 ^^. /\fy
= C^ e ' +0^6 cosl ^ T]*y + C e sin\^^ t]*; (19.^^6)
If J moreover, one measitreB the ti— coordinate from the inner houndary
point, thus puts:
j^* = j^* - Ti*^
the solution (equation (I9A6)) can also "be written:
Hi ... 21
=Tpf 2
F = d-| e + dp e cos
f ^) . Ija 2 =1. (f V
From the to-andary conditions (equation (I9.U2)) res'jlt for the constants
the values:
Tl* = 0.981 ; Tl* = - 2.01+ ; Tl* = - 3.02
d-L = -0.0062; d2 = O.987J d^ = 0.577
For the width of the mixing region one oh tains:
t = ^[\ - \) = ^^2c2 ^^* - n|^
h = 3.02X^/20 2 X
The constant c must he determined from experiments. From measurements
it is found that
h = 0.255 X (19.^7)
MCA TM No. 1218 65
Hence
^
2c^ = 0.08ij-5j c = 0.0174
and
^ = 0.0682 (19-i^8)
It is striking that here the ratio z/b is essentially smaller than for
the wake.
The distrihution of the velocity components u and y over the
width of the mixing zone is represented in figure 98.
From the second equation of motion one may calculate the pressure
difference "between the air at rest p ' and the homogeneous air stream p-, .
One finds:
p -p = o.ooJ+8 S u (19.49)
i o 20 \ ^ ^J
Thus an excess pressure of one— half percent is present in the Jet. For
the inflow velocity of the entrained air one finds according to
equation ( I9 . 40a ) :
^- CO = - FCrig) Uq = + 0.379 V 2c^ Uq
and with the measured value of c;
v_ „ = 0.032 Uq (19.49a)
d. The Plane Jet
In a similar manner one may also calculate the plane turbulent jet
flowing from a long narrow slot (compare fig. 94). The laws for the
increase of the width and the decrease of the center velocity have
already been given in equations (I9.7) and (I9.9): "b «v Xj Uj^ ~ rj=-
The calculation of the velocity distribution was carried out by Tollmien
(reference 8I); it leads to a non— linear differential equation of the
second order the integration of which is rather troublesome. Measurements
for this case were performed by Forthmann (reference 91)- In figure 99
66
MCA TM No. 1218
the measurements are compared with "t^he theoretical curTe. The agreement
Is rather good. Only in the neighhorhood of the velocity maximum is
there a slight systematic deviation. There the theoretical curve is
more pointed than the meas'Jired curve; the theoretical curve^ namely,
again has at the maximum a vanishing radius of curvature.
According to the Prandtl formula, equation (15.9)^ the exchange
hecomes zero at the velocity maximum, whereas actually a small exchange
is still taking place.
e. Connection Between Exchange of Momentum,
Heat and Material
In concluding the chapter on turlsulent flows I should like to
make a few remarks ahout the connection between the turbulent exchange
and the heat and material transfer in a turbulent flow.
In the Prandtl theorem equation (I5.9) for the apparent turbulent
stress:
T = p Z
one can interpret:
A
P I
Su
Bu _ . Su
kg sec I
m
(19.50)
as a mixing factor. It has the same dimension as the laminar viscosity [x.
Furthermore, the shearing stress t may be interpreted as a momentum flow:
momentum
T = = momentum flow
2
m sec
(19.51)
Momentum = mass X velocity = [kg sec] .
Another effect of the turbulent mixing phenomena, besides the
increased apparent viscosity by transport of momentijm,is the transport
of all properties inherent in flowing matter, as heat, concentration of
impurities, etc. If this concentration is not ■'jnlform, more heat or
imparity is carried away by the turbulent exchange from the places of
higher concentration than is brought back from the places of lower concen-
tration. Thus there results, on the average, transfer from the places of
higher to those of lower concentration.
NACA TM No. 12l8 6?
This results J for temperat-ure differences^ in a turliulent heat
transfer; for concentration differences (for instance, of salt), in a
turhiolent diffusion. They can, in analogy to eq^uation (I9.5O) be expressed
as follows:
,, , „-. momentum transport „ d / moment omN
Momentum flow = = A,- — I — —, 1
2 ' dv Vunit massy
m sec ''
Heat flow = heat transport = a^ ^ ( heat_\
2 Q dy \unit mass/
m sec
material N
„, ri . . -, transport of material . d /
Flow of material = = A, — ( ■
2 ^ dy V
cLy \unlt mass/
m sec
The heat content of the unit mass is -c-e ( = temperatiire.
c = specific heat
cal m
For chemical or mechanical
2
kg sec degree _
concentrations the concentration of material per unit mass is called the
concentration c; it is therefore the ratio of two masses and therefore
dimensionless. Thus the ahove equations may also be written in the
following forms:
T = A^ —
T^dy
> (19.52)
, ^- \ dy
The question arises as to whether A^, Aq, Aw are numerically the same
or different. If the momentum is transported exactly like heat or material
concentration — Prandtl's theorem is "based on this assumption — it would
follow that A^ = Aq = Aj, and, for instance, the velocity and temperature
distributions in a turbulent mixing region would have to be equal.
However, measurements show partially different behavior.
One has to distinguish between wall turbulence and free turbulence.
Concerning free turbulence, calculations of G-. I. Taylor (reference 92)
and measurements of Fage and Falkner (reference 93) showed for the velocity
and temperature profile of the plane wake flow
Aq
-^ = 2 (free turbulence) (I9.53)
At-
68 NACA TM No. 12l8
The heat exchange Is, therefore, larger than the momentum exchange.
Conseq^uently the temperature profile is wider than the Telocity profile.
The theory given for that phenomenon by G. I. Taylor operates with the
conception that the particles, in their turbulent exchange moTements,
do not maintain their momentum (Prandtl), hut their vortex strength 4^.
(Prandtl's momentum exchange theory »« Taylor's vorticity transfer
theory). However, there are cases not satisfied "by the Taylor theory
(for instance the case of the rotationally symmetrical wake). That the
heat exchange for free turbulence is considerably larger than the momentum
exchange is also shown by experiments of Gran Olsson (reference 88)
concerning the smoothing out of the temperature and velocity distributions
behind grids of heated rod's. With increasing distance behind the grid
the temperature differences even out much more rapidly than the differ-
ences in velocity.
For wall turbulence the difference between the mixing factors for
momentum and temperature is smaller. H. Eeichardt (reference 8?) was
able to show, from measurements of the temperature distribution in the
boundary layer on plates in longitudinal flow by Elias (reference 86)
and in pipes by H. Lorenz (reference Ik), that here
A
-^ = l.k to 1.5 (wall turbulence) (19. 5U)
T
Herewith we shall conclude the considerations of free turbulence.
CHAPTER XX: DETERMINATION OF THE PROFILE DRAG FROM
THE LOSS OF MOMENTUM
The method, previously discussed in chapter IX, of determining
the profile drag from the velocity distribution in the wake is rather
important for wind tunnel measurements as well as for flight tests; we
shall therefore treat it in somewhat more detail. The determination of
the drag by force measurements is too inaccurate for many cases, in the
wind tunnel for instance due to the large additional drag of the wire
suspension; in some cases (flight test) it is altogether impossible. In
these cases the determination of the drag from the wake offers the only
serviceable possibility.
The formula derived before in chapter IX, equation (9.^1) for
determination of the drag from the velocity distribution in the wake is
valid on.ly for relatively large distances behind the body. It had been
assumed that in the rear control plane (test plane) the static pressure
equals the pressure of the undistijrbed flow. However, in practically
carrying out such tests In the wind tunnel or in flight tests one is
NACA TM No. .1218
69
forced to approach the "body more closely. Then the static pressiire gives
rise to an additional term in the formula for the drag. For measurements
close behind the "body (for instance, for the wing, for x < t) this term
is of considerable importance, so that it .must he known rather accurately.
A formula was indicated, first hy Betz (reference 84), later hy B. M. Jones
(reference 85) which takes this correction into consideration. Although
at present most measurements are evaluated according to the simpler Jones
formula, we shall also discuss Betz' formula since its derivation in
particular is very interesting.
a. The Method of Betz
One imagines a control surface surrounding the Tsody as shown in
figure 100. In the entrance plane I ahead of the "body there is flow
with free— stream total pressure g^, hehind the "body in plane II, the
total pressijre gp < g . The lateral boundaries are to lie at so large
a distance from the "body that the flow there is undisturhed. In order
to satisfy the continuity condition for the control surface the velocity
U2 in plane II must he partially greater than the undisturhed velocity
Ur
Consider the plane prohlemj let the tody have the height h.
Application of the momentum theorem to the control surface gives:
W = h
n+ 00 «+ 00
(^1 ^ ^\> - (^
u — 00 J — 00
Pg + Pu 2 ' *^
(20.1)
In order to make this formula useful for test evaluation the integrals
must he transformed in such a manner that the Integrals need to he extended
only over the "wake". For the total pressures
A
at infinity:
in plane I:
In plane II:
O O d O
P 2
80 = Pi + 2 ^1
P 2
(20.2)
/
Outside of the wake the total pressure everywhere equals g^. Hence
equation (20.1) becomes
70
NACA TM Wo. 1218
W = li
1+ CO
(so - S2)
dy + ^
u, —
u^ ) dy
(20.3)
Thus the first Integral already has the desired fomij since the integrand
differs from zero only within the wake. In order to give the same form
to the second integral, one introduces a hypothetical substitute flow
u.2'(y) ^^ plane II which agrees with Ug everywhere outside of the wake,
Ug within the wake "by the fact that the total pressure
g . Thus
hut differs from
for u„' equals
+ ^u».
(20A)
Since the actual flow
Up satisfies the continuity equation the
^2_' "2
flow volume across section II for the hypothetical flow u.^ , u ' is too
large. It shows a source essentially at the location of the "body which
has the strength
Q = ^J (^* - ^2) ^
(20.5)
A source in a frictionless parallel flow experiences a forward thrust
E = - P U„ Q
(20.6)
One now again applies the momentum theorem according to equation (20. 3)
for the hypothetical flow with the velocity u at the cross section I
and the velocity u'p at the cross section II. Since go = gQ and the
resultant force, according to equation (20.6), equals R, one ohtains
- P TT,
oft = 2\
A
Tin
U
•2>
• (20.7)
By subtraction of equation (20.7) from equation (20. 3) there results
W + p Uq Q = h
/(So - S2)^y ^ 2/(^'2' - -2') ^y
(20.8)
NACA TM lo. 1218
71
or because of equations (20. 5) and (20.6):
W = h
/(go - S2)^y ^ |/("'2' - ^2')^ - P ^o /(-'2 - -2X
One may now perform each of these integrations only across the wake, since
outside of the wake u» = u . Due to u'g^ - ug^ = C u» - u Vu» + u )
a transformation gives the following formula:
(20.9)
Betz' Formula
In order to determine W according to this equation, one has to measure
in the test cross section "behind the "body the following values:
1. Total pressure gj (therewith g is the value of gp outside
of the wake).
2. Static pressure Pp.
Furthermore, p = static pressure at infinity.
Hence one obtains all quantities required for the evaluation of
equation (20.9 ).
It is useful for the evaluation of wind tunnel tests to introduce
dimenslonless quantities. With F = ht as area of reference for the
drag:
V = c^ h t I u/
and hence from equation (20.9)
(20.10)
For the case in which p = p =0, at the test cross section one can
2 o
write this equation, "because g^ = q^:
72
NACA TM No. 12l8
1 -
^^--JH-m
(20.11)
This agrees with equation (^.kl).* Thus in this case Betz' formula
changes, as was to he expected, into the previous simple formula.
h. The Method of Jones
Later B. M. Jones (reference 85) indicated a similar method which
in its derivation and final formula is somewhat simpler than Betz* method.
Let cross section II (fig. 101) (the test cross section) lie close
hehind the hody; there the static pressure p^ is still noticeably
different from the static pressure p^. Let cross section I he located
so far hehind the hody that the static pressure there equals the undis—
turhed static pressure. Then there applies for cross section I according
to equation (9.UI)*
W = h p
/ ^1 (Uq - ^1) ^1
(20.12)
In order to relate the value of u-, hack to measurements at cross
section II, continuity for a stream filament is first applied:
P ^1 iyn = P u„ dy^
(20.13)
* In chapter IX the total drag of the hody (both sides of the
plates) was designated by 2 Wj here the entire drag equals W!
NACA TM No. 12l8
73
Jones makes the further assumption that the flow from cross section II
to cross section I is without loss, that is, that the total pressure is
constant along each stream line from II to I:
So = Sn (20. Ill)
First, according to equations (20.12) and (20.13);
W = hp /u2(u^-ujdy.
(20.15)
Furthermore :
P 2
p+— U =g=q
■^o 2 o °o ,^o
Po + - ^■
P 2
2\ =h = ^2
P p
p + — u '^ = go
2 2 2
with p =0
o
> (20.16)
and hence
^2 \ I ^2 ~ P2
U
Un
U_
(20.17)
n P
From equation (20.12) follows, with W = c t h ^ U^ :
w
-Sf^
^l^ . ^2
^r~
and hecause of equation (20.1?)
(20.18)
74 NACA TM Wo. 1218
Formula of Jones
Thus all quantities may te measured in cross section II close to
the 'bod.j. This formula is simpler for the evaluation than Betz' formula,
equation (20.10).
In the limit, when the static pressure in the test cross section
"becomes P2 = P , this formula, of course, must also transform into the
simple formula equation (20.11). One obtains for po = p^ = from
equat i on ( 20 . I8 ) :
This is in agreement with equation (9.41). *•
Fifteenth Lecture (March I6, 19lf2)
CHAPTER ZXI: ORIGIN OF TURBULENCE
a. General Remarks
In this section a short summary of the theory of the origin of
turhulence will "be given. The experimental facts concerning laminar/
tur"bulent transition for the pipe flow and for the "boundary layer on the
flat plate have "been discussed in chapter XIII. The position of the tran-
sition point is extremely important for the drag pro"blem, for instance
for the friction drag of a wing, since the friction drag depends to a
great extent on the position of the transition point.
The so— called critical Reynolds nurriber determines transition. For
the pipe (ud/v) . , = 23OO, and for the "boundary layer on the plate
(Uqx/v) = 3 to 5 X 105. However, experimental investigations show
the value of the critical Reynolds numher is very dependent on the
initial distur'bance. The value of Re . , is the higher the smaller the
crit °
initial disturbance. For the pipe flow the magnitude of the initial
disturbance is given by the shape of the inlet, for the plate flow by
the degree of turbulence of the oncoming flow. For the pipe, for instance,
a critical Reynolds number (^<i/'^)crit ~ ^0,000 can be attained with very
special precautionary measures.
According to today's conception regarding the origin of turbulence,
transition is a stability phenomenon. The laminar flow in itself is a
solution of the Navier-Stokes differential equations up to arbitrarily
high Reynolds numbers. However, for large Ee-^iumbers the laailnar flow
NACA TM Wo. 1218 75
■becomes unetatle, in the sense that small chance disturhances (fluctu-
ations ,ln velocity) present in the flow increase with time and then alter
the entire character of the flow. This conception stems from Eeynolds
(reference 101). Accordingly, it ought to he posslhle to ohtain the
critical Eeynolds number from a stahility investigation of the laminar
flow.
Theoretical efforts to substantiate these assumptions of Eeynolds
mathematically reach rather far hack. Besides Eeynolds, Eayleigh
(reference 102) in particular worked on the problem. These theoretical
attempts did not meet with success for a long time, that is, no instability
could be established in the investigated laminar flows. Only very
recently has success been attained, for certain cases, in the theoretical
calculation of a critical Eeynolds number.
One assumtes for the theoretical investigations that upon the basic
flow which satisfies the ITavier— Stokes differential equations a disturbance
motion is superimposed. One then investigates whether the disturbance
movement vanishes again under the influence of friction or whether it
Increases with time and thus leads to ever growing deviations from the
basic flow. The following relations will be intorduced for the plane
case:
basic flows: U(x, y)^. Y(x, y); P(x, y)
disturbance movement: u'(x, y). v' (x, j) ; p'(x, y)
resultant movement: U + u* . Y + v' ; P + p'
> (21.1)
J
P, p* signify pressure. The investigation of the stability of such a
disturbed movement was carried out essentially according to two different
methods :
1. Calculation of the energy of the disturbance movement.
2. Calculation of the development of the disturbance movement with
time according to the method of small oscillations.
I am going to say only very little about the first method since it
was rather unsuccessful. The second method was considerably more
successful and will therefore be treated in more detail later.
The first method was elaborated mainly by H. A. Lorenz
(reference IO3). Thp following Integral expression may be derived for
the energy balance- of the disturbance movement:
^ / EdV = p / MdY - 11 / NdY (21.2)
16
WACA TM Wo. 1218
In it E = -^ (u' + t' ] signifies the kinetic energy of the distur'baiLce
movement. The Integration is performed over a space which participates
in the movement of the basic flow and at the "boundaries of which the
velocity equals zero, jrr signifies the substantial derivative. Thus
one finds on the left side of equation (21.2) the increase with time of
the energy of the disturbance movement. On the right slde^
M = -
u'
,2 SU
ax
,2 av
v' +
Sy
u'v« +
VSy
> (21.3)
N =
vSx 3y /
y
The first integral signifies the energy transfer from the main to the
secondary movement j the second the dissipation of the energy of the
secondary movement. If the right side is greater than zero, the intensity
of the secondary movement increases with time, and the basic flow is thus .
unstable. An assumed disturbance movement u', v* satisfies merely the
continuity equation, but no heed is paid to its compatibility with the
equations of motion. If one could prove that the right side is negative
for any arbitrao^y disturbance movement u*, v* this would serve as proof
of the stability of the basic flow. On the other hand, the instability
would be proved as soon as the right side is positive for a possible
disturbance. Unfortunately general investigations in this direction
are very difficult and have not led to much success. H. A. Lorenz
(reference IO3) treated as an example the Couette— flow (fig. 102), assuming
an elliptical vortex as a superimposed disturbance movement. He found for
/U d\
this case V — tt— / = 288, whereas Couette' s measurements for this case
^ V /fcrit '
gave the value I9OO.
b. The Method of Small Oscillations
For the second method (method of small oscillations) the disturbance
movement is actually calculated, that is, its dependance on the spatial
coordinates x, y and the time t is developed on the basis of the
hydrodynamic equations of motion. We shall explain this method of small
oscillations in the case of a plane flow. In view of the applications
of this method we shall immediately assume a special basic flow: the
component U, namely, is to be dependent only on y and t and Y = 0.
Such basic flows had been previously called "layer flows". They exist
for instance in tunnel flow and pipe flow, approximately, however, also
in the boundary layer since here the dependence of the velocity component
XJ on the longitudinal coordinate x is very much smaller than the
dependence on the transverse coordinate y. One now assumes a basic flow
NACA TM Wo. 1218
U(y, t); Y= Oj P(x, y)
77
(21A)
This basic, flow, ty itself, then satisfies the Navier-Stokes equations,
thus
^
Su 1 5p S u
— + = V —
Bt P Sx ^^2
Sy
=
(21.5)
/
A disturbance movement which is also two— dimensional is superimposed
upon this "basic flow:.
disturbance motion: u»(x,y,t); T»(x,y,t); p*(x,y,t) (21.6)
One then has as the
resultant motion: u=U+u'; t=0+t*; p=P+p' (21.7)
This resultant motion is required to satisfy the Navier— Stokes differential
equations and one investigates whether the disturbance motion dies away
or Increases with time. The selection of the initial values of the dis-
turbance motion is rather arbitrary, but it must of course satisfy the
continuity equation. The superimposed disturbances are assumed as "small",
in the sense that all quadratic terms of the disturbance components are
neglected relative to the linear terms. According to whether the dis-
turbance motion fades away or increases with time, the basic flow is called
stable or unstable.
By insertion in the Navier— Stokes differential equations (3.I8) one
obtains, neglecting the quadratic terms in the disturbance velocities
5U 5u» „ Su»
— + + U + V
St at Sx
. Su 1 Sp 1 Sp'
Sy P ax P Sx
+ A u'
— + u — -— i ^*
at ax p ay p ay
^ « V A V'
>(2l.8)
au» av«
ax ay
/
78
NACA TM No. 12l8
If one now notes the fact that the 'baBic flow by itself satisfies the
Navier— Stokes differential eguationsj equation (21.5) j equation (21.8)
is simplified to:
Su' TT ^' , SU 1 Sp' ^ ,
+ U + t' — + i— = V A u*
Bt Sdc Sy P Sx
St
^
P ay
SU« , SV« _ Q
-^
&vL+U^t1^1Se1^VAt» >
(21.9)
The pertinent "boundary conditions are: Tanishing of the disturbance
components u* and t' on the bounding walls. From the system
equation (21. 9) of three equations with three unknown quantities u*,
t', p» one may at first eliminate p' by differentiating the first
equation with respect to y and the second with respect to x and then
subtracting the second from- the first. This gives, with continuity taken
into consideration:
S^u' ^ jj B^u' + ^, ^ _ 5^' _ jj SV
BtBy
+ V
SxSy
a^u
,^ ^y
ay^
StSx
» S^u' S^t'
Sy SxSy
ax-^
Sx-
>
> (21.10)
/
In addition to this there is the continuity equation (21.9). There are
now two equations with two unknown quantities u*, v*.
Form of the Disturbance Movement
For cases where the basic flow predominantly flows in one direction
as for instance boundary— layer or pipe flow, the disturbance motion is
assumed to be a wave progressing in the x-direction (= main flow direction),
the amplitude of which depends solely on y. The continuity equation of
the disturbance motion may in general be integrated by a disturbance
function for which the following expression may be used:
NACA TM No. 12l8
19
* (x^y^t) = cp(y)e
ia(x-ct)
(21.11)*
Where:
X = 2it/a the waTe length of the disturbance (a = real)
C = C + 1 c.
r 1
c = Telocity of wave propagation
c^ = amplification factor; c . < 0: stable; c. > 0: unstable
9(y) = 9 (y) + i 9.(y) = amplitude of the disturbance movement
r 1
From equation (21.11) one obtains for the components of the distirrbance
movement
li' = T^ = 9'(y)e
oy
St , , , ia(x^t)
T» =-—- = - ioo:p(y)e
dx
> (21.13)
/
By substitution into equation (21.10) one obtains the following differ-
ential equation for the disturbance amplitude cp:
/ i
iafUcp" — ccp" - cpU" + a ccp - Ua cp
j = v^cp"" - 2a cp" + a 9 j
or
(U - c)(9" - a29) - u"q) = =^-i^ (cp"" - 2aV + o-V) (21. U)
Ow
■One introduces dlmensionless quantities into this equation by referring
all velocities to the maximum velocity U of the basic flow (that is
m
for the friction layer the potential flow outside of the boundary layer
and all lengths to a suitable reference length §) (for Instance, for
^he convenient complex formulation is used here. The real part of
the flow function, which alone has physical significance, is therefore
Re(t)
C-it
cp COS
r
a
(- - V)
cp. sin
1
a ( X —
c t
r ,
(21.12)
80
NACA TM Wo. 1218
the "boundary layer flow, the "bo-undary layer thickness). Furthermore,
differentiation with respect to the dimensionless quantity y/5 will
"be designated ty a prime mark (').
One then obtains from equation (21.1i|)
(U - c)(cp" - a^qp) - U"cp = -^ (q)"" - 2aV + a\)
(21.15)
DISTURBADJCE Dlb'Jb'KHEETIAL EQUATION
where E = ■■ ™ . This is the disturbance differential equation for the
amplitude cp of the disturbance movement. The boundary conditions are,
for instance, for a boiuidary layer flow
y = (wall): u' = v' = 0: 9 = cp' =
y = 00 : u' = "v* = 0: cp = qp' =
(21.16)
The stability investigation is an eigenvalue problem of this differential
equation for the disturbance amplitude <p(y) in the following sense: A
basic flow U(y) is prescribed which satisfies the Navier-Stokes differ-
ential equations. Also prescribed is the Eeynolda number R of the basic
flow and the reciprocal wave length a = 2it/x, of the disturbance movement.
From the differential equation (21. I5) with the boundary conditions
equation (21. I6) the eigenvalue c = Cp + i Cj_ is to be determined. The
sign of the imaginary part of this characteristic value determines the
stability of the basic flow. For Cj^ < the particular flow (U, R)
is, for the particular disturbance a, stable; for Cj^ > 0_, unstable.
The case Cj^ = gives the neutrally stable disturbances. One can
represent the result of the stability calculation for an assumed basic
flow U(y) in an a, R— plane in such a manner that a pair of values
Cj., c± belongs to each point of the a, R— plane. In particular the
: = in the a, R— plane separates the stable from the unstable
curve
disturbances. It is called the neutral stability curve (fig. IO3). In
view of the test results one expects only stable disturbances to be present
at small Reynolds numbers for all wave lengths a, unstable disturbances,
however, for at least a few a at large Reynolds numbers. The tangent
to the neutral stability curve parallel to the a-^axis gives the critical
Reynolds number of the respective basic flow (fig. IO3).
WACA TM No. 1218 81
Methods of Solution and Gteneral Properties of the Dlstiirban ce
Differential Equation
Since the stability limit (Cj^ = 0) is expected to occur at large
Reynolds numbers^ it suggests itself to suppress the friction terms in
the general disturhance differential equation and to ohtain approximate
solutions from the so— called frlctionless disturbance differential
equation which reads
(U - c) (cp" - a^q)) -U"q)
(21.17)
Only two of the four "boundary conditions^ equation (21.l6),of the complete
disturbance differential equation can now be satisfied since the friction-
less disturbance differential equation is of the second order. The
remaining boundary conditions are:
y = 0: v« = Oj 9 = Oj y = 00: t» = 0: cp = (21. 18)
The cancellation of the friction terms in the disturbance differential
equation is very serious, because the order of the differential equation
is thereby lowered from ij- to 2 and thus Important properties of the
general solution of the disturbance differential equation of the fourth
order possibly are lost. (Compare the previous considerations in
chapter IV concerning the transition from the Navler— Stokes differential
equations to potential flow. )
An Important special solution of equation (21.17) is the one for
a constant basic flow, U = constant^ which is needed for Instance for
the stability investigation of a boundary layer flow as a Joining
solution for an outer potential flow. One obtains from equation (21.17)
for
U = constant: cp = e
iay
However, due to the boundary conditions for cp at y = ", the only
permissible solution Is
cp = e^ y (21.19)
We shall prove at first two general theorems of Eaylelgh on the
neutral and unstable oscillations of the frlctionless disturbance
differential equation.
82 WACA TM No. 12l8
Theorem I: The wave Telocity c for a velocity profile with
U"(y) — must, for a neutral oscillation (c^ =0, c = c ), egual
the hasic velocity at a point so that there exists within the flow a
point U — c = 0.
Proof: (indirect) One makes the assumption c > U (= maximum,
velocity of the hasic flow). One then forms from equation (21.17) the
following differential expressions:
2 TT"
L(cp) = cp" - a cp qp = (21.20a)
U — c
and
2- U" —
L(cp) = cp" - a cp ^ cp = (21.201))
U - c
L(q)) signifies the expression o"btained from 1(9) , if one inserts
everywhere the conjugate complex q^uantities. Because of the "boundary
conditions
y = 0: cp = cp =
y=oo. q3 = q) = o
One forms further the expression 9L(cp) + cp L(cp) and integrates
it "between the limits y = and y = co. The integrals may "be taien up to
y = 00, since for large y, 9 »« e~^y. Because of equation (21.20aj b)
J2_ must then "be
■^1 =
cp L(q)) + cp L(cp)
dy = (21.21)
^y=o
After insertion of equation (21.20a and "b) results, "because c = c,
J = j ( cp" cp" + cp"(p - 2a^cp9 - 2 — - — cp9 j dy =
'y=o
or
WACA TM Wo. 1218
83
noo
J =
1
qxp' + cp'9
- 2
-"o
llo
cp'cp»dy - 2 I [ a^ + — ^^ ] cp9 dy =
U - c
The first term vanishes due to the houndary conditions, hence there
remains
Jl =-2
Jy=o
cp»cp' +( a^ + — 2 — jqxp
U - c /
dy =
(21.22)
U" -
cp'cp' as well as qxp are positive throughout; if U" — and c > U ,
U"/U' — c — and hence the integrand in eq^uation (21.22) is positive
throughout. Thus the integral cannot "become zero. The assumption made
at the "beginning c > U therefore leads to a contradiction.
For "basic flows with "D" — 0, as for instance "boundary— layer flows
in a pressure drop, the wave propagation velocity therefore must "be smaller
than U^ for neutral disturbances. Hence a point IT — c = exists
within the flow. This point is a singular point of the frictionless
disturbance differential equation (21.17) and plays as such a special role
for the investigation of this differential equation. The wall distance y
at which U — c = is called j = Y-y- = critical layer.
This first Rayleigh theorem proved a'bove applies — as shall "be noted
here without proof — in the same manner to flows with tT" > 0.
Sixteenth Lecture (March 23, 19^2)
Theorem II: A necessary condition for the presence of amplified
oscillations (c^ > 0) is the presence of an inflection point within the
"basic flow (U" = 0).
Proof: (indirect) According to assumption, c. ^ 0; thus
U - c / for all y. With L(cp) and L(cp) one forms, according to
equation (21.20a) and (21.20h), a simdlar expression as "before. This
latter, integrated from y = to y = °°, must again give 0, thus
noo
J2 =
[
9 L(q)) - cp L(cp) dy =
]■
y=0
(21.23)
8U
NACA TM No. 12l8
By substitution according to equation (21.20a and b) results with
c = c — 1 c.
r 1
'2 =
'7=0
cpcp" - cptp" - ^(—^ 21.
VU - c U - c
dy =
or
'2 =
W - qxp''
- 2i c.
JQ
u"
cpcp dy =
U - c
{2±.2k)
The first term again vanishes because of the boundary conditions. Since cp^
is positive thrcaghout and lu — c| / 0, the integral can only vanish if
U" changes its sign, that is, an inflection point of the velocity profile
U" = must be present within the flow. It has, therefore, been proved:
In order to make the presence of amplified oscillations possible, an ■
inflection point mast exist in the velocity profile of the basic flow, or,
expressed briefly, such oscillations are possible only for inflection
point profiles.
Later on Toll mi en (reference 110) proved that the presence of an
inflection point is not only a necessary but also a sufficient condition
for the existence of amplified oscillations. Hence the following simple
statement is valid: Inflection point profiles are unstable. It must be
mentioned that all these considerations apply in the limiting case B— ^<»
since the proofs were obtained from the frictionless disturbance differ-
ential eq_uation.
¥e know from our previous considerations about the laminar boundary
layer that inflection point profiles always exist in the region of pressure
rise, whereas in the pressure drop region the boundary layer profiles are
always without an inflection point, (fig. lOij-). Hence we recognize that
the pressure rise or pressure drop is of decisive significance for the
stability of a boundary layer flow.
The converse of the theorem just set up is also valid, namely, that
for E — >oo velocity profiles without inflection point are always stable.
From this, however, "one must not conclude that profiles without inflection
point are stable for all Reynolds numbers. A closer investigation for
Reynolds numbers of finite magnitude shows that there profiles without an
inflection point also become unstable. One is faced with the peculiar
fact that the transition from Re = » to Ee-number of finite magnitude,
that is, the addition of a small viscosity to a frictionless flow, has a
NACA TM No. 12l8 85
destabilizing effect, whereas one intuitively expects the opposite. As
later considerations will show in more detail, the typical difference
between the neutral stability curves of a basic flow with and without
inflection point appears as represented in figure IO5. For the velocity
profile without an inflection point the lower and the upper branch of the
neutral curve have , for E — > 00 , the same asymptote a = . For the
velocity profile with inflection point the lower and upper branch of the
neutral curve have, for E — >», different asymptotes so that for E = 00
a certain wave length region of unstable disturbances exists. Furthermore
the critical Eeynolds number is smaller for velocity profiles with an
inflection point than for those without an inflection point.
Hence it is to be expected for very large Ee-iiumber, to a first,
very rough approximation, that the transition point in the boundary layer
of a body lies at the pressure minimum. Figure IO6 shows schematically
the pressure distribution for a rather strongly cambered wing profile at
a small lift coefficient. The transition point would be expected in this
case Just behind the nose on the pressure side, slightly more toward the
rear on the suction side.
Solution of the Disturbance Differential Equation
In order to perform the actual calculation for the boundary— value
problem Just formulated, one needs at first a fundamental system
Cp-,, cp. °^ '^^^ general disturbance differential equation (21.15).
One Imagines the basic flow U(y) given in the form of a power series
development :
U "
U(y) = U' y + -^ y2 + . . . . (21.24)
° 2!
If one introduces this expression into equation (21.15) and. then wants
to construct a solution from the complete differential equation which
satisfies the boundary conditions (equation (21. 16)), one encounters
extreme difficulties of calculation, due to the two conditions to be
satisfied for y = 00, In order to obtain any solution at all, one has
to make various simplifications. The simplificationB concern:
1. The basic flow: Instead of the general Taylor— series
equation (21.2^4-) one takes only a few terms, thus for instance a linear
or a quadratic velocity distribution.
2. The disturbance differential equation: For calculation of the
particular solutions the disturbance differential equation Is considerably
simplified.
Eegarding 1, it should be noted that linear velocity distributions
frequently have been investigated with reepect to stability, as for
86 NACA TM No. 12l8
instance the Couette flow according to figiore 102 or a polygonal approxi-
mation for cuTTed Telocity profiles according to figure 107. This facili-
tates the calculation due to the fact that then the singular point
U — c = is avoided in the frictionless dist'ur'bance differential
equation (21.17) for neutral disturhances. However, all InvestigationB
with linear velocity diBtritutions (references 104, 105, 106) were unsuc-
cessful with the frictionless as well as with the complete differential
equation. No critical Reynolds number resulted. When one later took
for a "basis paraholic profiles, these negative results "became intelligl"ble.
One must, therefore, take at least a parahollc distrl'bution as a "basis
for the "basic flow.
Regarding 2, it should "be noted that one can provide approximate
solutions for the solutions of the complete differential equation (21.15)
from the frictionless differential equation (21.17) since the solutions
are required only for large Ee-^iumber E. The frictionless differential
equation however can yield no more than two particular solutionsj two
more have to "be calculated, taking the largest friction terms in
equation (21.15) into consideration.
The course of the calculation for the particular solutions will "be
"briefly Indicated. One limits oneself to neutral dlstjir'bances, assumes
a para'bolic velocity distrl'bution^ and imagines the latter developed in
the neighborhood of the critical layer.
y = yg.: U-c=U-c^ =
U - c = U'^(y - y^) . !^ (y - j^f (21.25)
The first pair of solutions 9-, j "Pp ^^ then o"btained from the friction-
less dlstur"bance differential equation (21.17) "by suhstltution of
equation (21.25). According to known theorems a"bout linear differential
equations with a singular point a linearly independent pair of solutions
has the form
^i=(^-^k)^(^-^e)
u"
(21.26)
P, and P are power series with a constant term different from zero.
The parttc-ular solution cp is especially interesting.
NACA TM No. 12l8
87
9',
-> °° for
J = It,
That Isj the u'— component of the dleturhance velocity becomes infinitely
large in the critical layer. This can also he understood directly from
the frictionlesB disturhance differential eq^uation (21.17). According
to equation (21.17)
cp" — a cp =
U"
U - c
9
or
>p"
j-^; ^' ~ ic8 (y - y^). If u"^ ^
This singular hehavior of the solution cp2 ^^ "the critical layer stems
of course from neglecting the friction. The frictionless differential
eq^uation here no longer gives a serviceable approximation. In the
neighborhood of the critical layer the friction must be taken into
consideration. Moreover, there is another inconvenience connected with
the cpp' For fulfillment of the boundary conditions one req^uires the
solution for y — y-g- > as well as for y — y„ < 0. However, for qpo
it is at first undetermined what branch of the logarithm, should be chosen
at transition from y — yg; ^ to y — y;g; "^ 0. This also can be clarified
only if in the neighborhood of y — y^-, at least, the large friction terms
of the complete differential equation (21.15) are taken into consideration.
The details of the calculation will not be discussed here. The calcu-
lation leads, as Tollmien (reference 109) tias shown, to the result that
one obtains for the solution cp2 a so— called transition— substitution in
the critical layer which appears as follows:
U"
7-J^>0: cpg = P2(y -J^) + —(7- ^k) ^l(y - ^k) ^°e (7 - Y^
u"
y - Jk < 0= ^2 = P2(y - ^k) ^ ^ (y - ^k) ^i(y - ^k) i°8 "
y - y-
K
- in
(21.27) '
If one writes, according to this, the complete u'— component, then in the
neighborhood of 7 ~ J-^'-
KACA TM No. 12l8
J - J^> 0: u' =
< 0: u«
U".
K
+ — 1'°S (j - J-^) COB (ox - pt)
.5^ .OS
u"
K
cos (ax - pt) S (21.28)
+ It sin (ax — pt)
y
One ottalns therefore in the critical layer a phase discontinuity for the
u* -component. This is retained eren in going to the limit, E — > 00 . it
is lost, however, if one neglects the curvature of the basic flow U"
or if one operates only with the frictionless differential equation. This
phase discontinuity is very significant for the development of the motion.
The loss of the phase discontinuity is the reason that stahility investi-
gations neglecting the curvature U" or operating only with the frictionless
differential equation remain unsuccessful.
With this friction correction in the critical layer the pair of
solutions cp
9.
is sufficiently determined. By taking the friction
'1' -^2
terms in equation (21.15) into consideration, one then obtains a second
pair of solutions 9 , cp, which can he represented hy Hanlcel and Bessel
functions. Of these -two solutions cp, tends very strongly towards
infinity and is therefore not used "because of the "boundary conditions,
equation (21. 16). cp tends, for large y, towards zero.
The Boundary Yalue Pro"blem
The general solution as a linear combination of the four particular
solutions is:
cp = C^cp^ +
CgiPg + C^cp^ + C^cpj^
(21.29)
Let us consider in particular the case where a "boundary layer profile is
investigated with r-espect to sta"bility. For this case the "boundary value
pro"blem can "be somewhat simplified. The previous considerations showed
that in the disturbance differential equation the friction essentially
needs to "be taken into consideration only in the neigh'borhood of the
critical layer; also, of course, at the wall, because of non— slip. The
critical layer is always rather close to the wall; hence for y > 6,
NACA TM Wo. 1218
89
where U = U^j^ = constant^ one may use the frictionlees solution which is
according to equation (21. I9) 9= e~°''^. Thus the condition that the
solution for y = 5 Joins the solution for U = constant is
9' + a cp = $ =
o 00
(21.30)
This mixed "boundary condition is therefore to he set up on the outer edge.
Furthermore, the particular solution cpi is a priori eliminated in the
general solution (equation (21.29)), since it grows, for positive y — jy-,
"beyond all limits j thus C. = 0. Hence there remains for the houndary
•value according to equation (21.16)
Ccp +Cq) +Cq) =0
1 lo 2 2o 3 30
C cp» + C cp' + C cp» =0
1 Id 2 2o 3 30
C$ +C<i) +C$ =0
ri5 2 26 3^35
~\
>
(21.31)
y
A further simplification takes place "because of the fact that "because of
the rapid fading away of the solution cp on the outer edge y = 8, the
solution $_ already practically equals zero. In the third equation of
15
equation (21. 3I)
35
may therefore he cancelled. The boundary value
problem actually to he solved is, therefore.
qp cp cp
^lo ^2o 30
cp* cp' cp*
^ lo ^ 2o ^ 2o
15
$^0
25
=
(21.32)
This determinant gives the eigenvalue problem indicated above, which
requires - as has been said before - the solution of the following problem:
Given
90 NACA TM No. 12l8
1. "basic flow U(y)
2. Eeynolds number Ee = U S/v
3. wave length of the disturhance a = 23t/a.
One seeks from equation (21.32) the pertinent complex eigenvalue
+ 1 c . . Therein c gives the velocity of wave propagation and
c. the anrplification or damping.
0=0
r
Equation (21.32) may formally "be written in the foirTa:
F(a, c^, c., E; TJ'^, U"^, ....)= (21.33)
where equation (21.33) signifies a complex equation^ hence is equivalent
to two real equations
f fa, c , c., E; U' , U" , . . . =
IV r 1 o o '
fJa, , c, E; U» , U" , . . . )
2\~' r' 3.' ^ o-* o-* /
> (21.3^)
If one imagines for Instance Cp eliminated from these two equations,
one obtains one equation "between a, E, c. :
gJa,, c^, E; U»Q, U"^, . . J) = (21.35)
From this equation Cj_ can be calculated as a function of a and E.
The constants U*o, U"o, • • • are parameters of the basic flow. Thus,
if equation (21.35) is assumed solved with respect to c..
i = gg^a, E; U'q, U"o, . . .j (21. 36)
Cj =
Finally one obtains from it, for the neutral disturbances c. = 0, a
curve in the a, E— plane, given by the equation
gg/a, E; U»^, ir^, . . .) (21.37)
WACA TM No. 1218 91
This is the sought for neutral stahllity curve (compare figure IO3), which
separates the unstable from the statle disturhances and also yields
the theoretical stahility limit, that is, the critical Eeynolds number
^^crif
The performance of the calculation, here only indicated, is analyt-
ically not possible since the quantities a, c , E enter into the
determinant, equation (21. 32), in a very complicated manner. One has
therefore to resort to numerical and graphical methods. The critical
Eeynolds number is Tery largely dependent on the form of the velocity
profile of the basic flow, in particular on whether the velocity profile
of the basic flow has an inflection point, thus on U"(y).
The critical Eeynolds number found from such a calculation gives
exactly the boundary between stability and instability, hence the first
occurrence of a neutrally stable disturbance. In comparison with the
transition point of test results it is therefore to be expected that the
experimental transition point appears only for larger Eeynolds numbers
where an amplification of the unstable disturbance has already occurred.
c. Eesults
A few results of such stability calculations will be given. The
completely calculated example concerns the boundary layer on the flat
plate in longitudinal flow with the laminar velocity profile according
to Blasius (compare chapter IXa). In figure IO8 the streamline pattern
of this plate boundary layer with the superimposed disturbance movement
is given for a special neutral distvirbance. Figure IO9 shows, for the
same neutral disturbance, the amplitude distribution and the energy
balance. Since the distizrbance in question is neutral, the energy
transfer from the main to the secondary movement is of exactly the same
magnitude as the dissipation of the energy of the secondary movement.
Figure 1^0 shows the neutral— stability curve as result of the stability
calculation according to which the critical Eeynolds number is referred
to the displacement thickness b* of the boundary layer
(Uj^S'**'/v )„pj+ = 575- The connection between displacement thickness S*
and length of run x is for the laminar boundary layer according to
equation (9. 21)
V*
= 1.73
Thus a critical Eeynolds number formed with the length of run
(U x/v) .^ = 1.1 X lo5 corresponds to the critical Ee-number
(U 5*/v) = 575. The critical number for this case observed in tests
^ ' crit
92 NACA TM No. 12l8
was 3 to 5x10-^. It was explained atove tJaat it must be larger than
the theoretical number. Furthermore^ figure 110 shows that at the
stability limit the unstable wave lengths are of the order of magnitude
X = 55- The unstable disturbances thus have rather long wave lengths.
This calculation, carried out by Tollmlen (reference IO9) for the
flow without pressure gradient was later applied by Schlichtlng
(reference 11^4-, 115) "to boundary layer flows with pressure drop and
pressure rise. The boundary layer profiles with pressure rise and
pressure drop can be represented in a manner appropriate for the stability
calculation as a one— parameter family with the form parameter X-p i^
according to Pohlhaueen's approximate calculation. One then obtains for
each profile of this family a neutral— stability curve as Indicated in
figure 111. Hence the critical Eeynolds number (^r^* / '^) crit ^^ ^
function of the form parameter X-p h according to figure 112. In
retarded flow (X-p r <0) the critical Ee-number is smaller than for the
plate flow (^pi^ = 0) , for accelerated flow {'>^-pu ^ 0) it is larger.
With this result of a universal stability calculation the position of the
theoretical transition point may be determined conveniently for an
arbitrary body shape (plane problem) in the following manner: At first,
one has to calculate for this body the potential flow along the contour,
ftirthermore one has to carry out, with this potential flow, a boundary
layer calculation according to the Pohlhausen method. This calculation
yields the displacement thickness and the form parameter Xp^, as
functions of the arc length along the contour, in the form
- f^(s) and Xp^ = f2(B)
Since in general there exists, accelerated flow at the front of the body
and retarded flow at the rear X-nj. decreases from the front toward the
rear. By means of the universal stability calculation according to
figure 112 one may determine a critical Reynolds number i^-n^* / "^^ crit
for each point of the contour. The position of the transition point
for a prescribed Ee-^umber U^t/v is then given by the condition
^ ^crlf V V V /crit
Figure 113 shows, for the example of an elliptic cylinder*, how to find
the transition point. The curve (Uj^^-^/v)^^^^ decreases from the front
*The boundary layer calculation for this elliptic cylinder was given
in figure 52. ,
NACA TM No. 12l8 93
toward the rear; the curve Ujj^S^/v for a fixed Ee-^aumber U t/v increases
from the front toward the rear. The intersection of the two curves gives
the theoretical transition point for the respective Ee— number U-t/v.
By determining this point of intersection. for various U-t/v one ohtains
the transition point as a function of U t/v. The result is represented
in figure 114. The transition point travels with increasing Ee-durober
from the rear toward the front; however, the travel is considerahly
smaller than for the plate in longitudinal flow which is represented in
figure llij- for comparison. Finally figure II5 shows the result of such a
stability calculation for four different elliptic cylinders in flow
parallel to the major axis. The shifting of the transition point with
the Ee— number increases with the slenderness of the cylinder. For the
circular cylinder the shifting is very slight, which is caused hy the
strongly marked velocity maximum. As a last result, figure II6 shows the
travel with Ee— number of the transition point on a wing profile for various
lift coefficients. The profile in question is a symmetrical Joukoweky
profile with lift coefficients Cg^ = to 1. With increasing angle of
attack the transition point travels, for fixed Ee-nijmber toward the front
on the suction side, toward the rear on the pressure side. (compare the
velocity distirbutions for this profile, given in figure 5^.) One recog-
nizes that the shift of the transition point with the lift coefficient
is essentially determined by the shift of the velocity maximum.
The last examples have shown that it is possible to calculate
beforehand the position of the transition point as a function of the
Ee— number and the lift coefficient for the plane problem of an arbitrary
body immersed in a flow (particularly a wing). Eegarding the comparison
with test results it was determined that the experimental transition
point always lies somewhat further downstream than the theoretical tran-
sition point. The reason is that between the theoretical and the experi-
mental transition points lies the region of amplification of the unstable
disturbances. This amplification also can be calculated on principle
according to methods similar to those previously described. (Compare
Schllchting (reference 112) where this was done for the special case of
the plate in longitudinal flow. ) Presumably one can obtain a still closer
connection between the theoretical instability point and the experimental
transition point by applying such calculations to the accelerated and
retarded flow.
CHAPTER XXII. CONCERNING THE CALCULATION OF THE TURBULENT FRICTION
LAYER ACCORDING TO THE METHOD OF GEUSCHWTTZ (REFERENCE 78)
a. Integration of the Differential Equation of the
Turbulent Boundary Layer
In order to integrate the system of equations (I8.I6), one
first introduces dimensionless variables. One refers the lengths to the
9k
NACA TM No. 12l8
wing chord t and the velocity to the free stream velocity U^, thus:
t "" '
t
S
1^:
= c* =
IP
(22.1)
Hence the system of equations (equation (I8.I6)) may "be written:
— + o.oo89i^ -^ = 0.001^61 f-^'f i-
dx* t3* \U J -d*
djc* \ 2/ U. dx*
o
(22.2)
First, the second equation is solved with constant values for ''"o/pu
and H, namely
^
= 0.002; H = 1.5
The first approxinjation '"3 *(x*) obtained from that is then substituted
in the first differential equation. From the latter one obtains a first
approximation ^-,*(x*) and from that, according to equation (22.1),
Ti-|(x*). With T], (x*) one determines according to figure 92 the course
of H(t)) and corrects t according to equation (18.I5). Then one
obtains from the second differential equation a second approximation
For the solution of the differential equations one uses the isocline
method which can be applied for the present case, according to Czuber,
in the following manner: Both differential equations have the form:
g + f (x)y = g(x)
(22.3)
As can be easily shown, this differential equation has the property that
all line elements on a straight line x = constant radiate from one point.
The coordinates of this point (= pole) are:
NACA TM No. 12l8 95
i = X + -i-; X = 44 (22A)
f(x)' f(x)
Thus one has only to calculate a sufficient numiber of these poles and
can then easily draw the integral curve.
Figure 93 indicates the result of such a calculation for the profile
J 015; °a ~ ^' '^^^ calculation of the laminar boundary layer for the
same profile was performed in chapter XII, figure 49 ^ tahle 6.
Initial values: The transition point was placed somewhat arhitrarily
at the velocity maximum of the potential flow (x/t = 0.l4l). It was
assumed that:
Ee = -2_ = 10
V
For the laminar "boundary layer was found:
U t
^ = 0.141: ^ 1 -^ = 1.56 (tahle 6)
Hence there results, with S*/i3 = 2.55;
- = O.lU: (1) = 0.611 X 10-3 (tahle 8)
The corresponding r] — value was ass'jmed to be
T] = 0.1 (table 8)
Calculation to the second approximation suffices. The result is compiled
in table 8 and figure 93. A turbulent separation point does not exist
since r\ remains below 0.8. From the variation of the shearing
stress T along the wing chord the drag coefficient of the surface
friction may be determined:
W=2b| T dx (x= measured along chord)
96
-^kCk TM No. 1218
or c,^ = W/2 Id t § Uc
w
°w =
(22.5)
The e-valuation of the Integral gives
c^ = 0.0090
t. Coimection Between the Form Parameters ti and H = tS/S*
of the Tiirhiilent Boundary Layer
According to Pretsch (reference 80) one may also represent analyti-
cally the relation "between the form parameters t] = 1 — (u^/tr)^ and
H = 5*/t3 which was found empirically by G-ruschwitz, compare figure 92.
A power law is set up for the velocity distribution, of the form:
H -f-L^ _ n
U 1^8
;
= z
(22.6)
with n = 1/6, 1/1, 1/8 . .
Hence results:
according to the experiments so far.
5*
6
ni
1 -
n\ n
z dz =
n + 1
(22.7)
dy/5=0
Furthermore ;
1
5
0^1= ^"(^-^")
s^-f)^l=l ^-
dz
(22.8)
n _ 7,2n
z^^ - z
i\z = _J^ L_ = S
y n+1 2n+l (n+ l)(2n + 1
z=0
NACA TM No. 12l8
97
From equations (22.7) and (22.8) follows:
H = ii- = 2n + 1
or
n =
H - 1
(22.9)
From equation (22.8) follows further
i = H - 1
5 H(E + 1)
(22.10)
The GruBchwitz form parameter t) Is defined according to equation (l8.2)
hy:
'u
Tl = 1 -
■■3\2
U,
With equation (22.6) t] becomes:
1 -
iT
(22.11)
Substitution of equation (22.10) into (22.11) gives;
(22.12)
The connection "between H and t\ calculated according to this equation
is given in the following table and is also plotted In figure 92. The
curve calculated according to equation (22.12) almost coincides with the
curve found empirically by Gruschwltz.
98
NACA TM No. 12l8
H
T\
1
1.1
0.270
1.2
O.i^Oi^
1.4
0.573
1.6
0.688
1.8
0.772
2.0
0.833
2.2
0.881
2.k
0.916
2.6
0.941
2.8
0.959
3.0
0.972
Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics
WACA TM Wo. 1218 99
BIBLIOGRAPBY
Tart B. — Tiirtulent Flow
Chapters XIII to XX
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61. Reynolds, OslDorn: Phil. Trans. Eoy. Soc, I883 or Collected papers
vol. II, p. 51*
62. Prandtl, L.: Ueter den Luftwlder stand von Ku^eln, Gottingen
Nachrichten, p. I77, 19114-.
63. Lorenz, H. A.: Abhandlungen ub. theoretische Physik. Bd. 1,
pp. k3 - hi, 1907.
6k. Boussinesq, T. V.: Mem. Pres. par div. Sav., vol XXIII, Paris, I877.
Theorie des I'ecoulement tourtillant, Paris I987.
65. Eeichardt, H. : Mess^jngen turtulenter Schwankungen . Naturw. I938,
p. h-Ok.
66. Prandtl, L. : Ueber die ausgebildete Turtulenz. ZAMM I925, p. I36 and
Verhdl. II. intemat. Kongress f. angew. Mech. Zurich, I926.
67. Schmidt, W. : Massenaustausch in freier Atmophare und verwandte
Erscheinungen. Hamburg, I925.
68. V. Karman, Th. : Mechanlsche Aehnlichkeit und Turtulenz. Nachr. Ges.
Wiss. Gottingen, Math. Ehys. Elasse, p. 58, I93O.
69. Blasius, H. : Forschungsheft I3I des Tereins deutscher Ingen. . I9II.
70. Nituradse, J.: Stromung in glatten Rohren. "VDI— Forschungsheft 356,
1932.
71. Nikuradse, J. : Stromungsgesetze in rauhen Rohren VDI— Forschungsheft
361, Berlin 1933-
72. Schlichting, H. : Experimentelle XJntersuchungen zum Eauhigkeitsproblem.
liig.-Arch. Bd. 7, I936, p. 1.
73. Prandtl, L.: Ueber den Re ibungswlder stand stromender Luft. Results
of the Aerodynamic Test Institute, Gottingen, III. Lleferung,
1927.
jh. Prandtl, L. : Zur turhulenten Stromung in Rohren und Langs Platten.
Restilts of the Aerodynamic Test Institute, Gottingen, IT.
Lieferimg, 1932.
■"The reference numbers used herein have "been taken directly from
the original German Version.
100 NACA TM No. 12l8
75. v. Karmaxi, Th. : Verhdlg. III. Internat. Mech. Kongreas, Stockholm,
1930.
76. Prandtl, L., and Schlichting, H. : Das Widerstandsgesetz rauher Platten.
Werft, Eeederei, Hafen. 1934, p. 1.
77. Tanij Hama, Mituiai: On the Permissible Eoughness in the Laminar
Boundary Layer. Aeron Bes. Inst. Tokyo Eep. I5, U19, I9U0.
78. G-ruBchwitz, E. : Die turbulente Eeibungsschicht in etener Stromung
bei Druckabfall und Druckanstieg. Ing.-Arch. Bd.II, p. 321, I93I.
79- Millikan, C B.: The Boundary Layer and Skin Friction for figure of
EeTolution. Trans. Americ. Soc. Mech. Eng., vol. 54, No. 2, 1932.
80. Pretsch, J. : Zur theoretischen Berechnung des Profilwlderstandes.
Jb. 1938 d. deutschen Luftfahrtforshung, p. I 6I.
81. Toll mi en, W. : Berechnung der turbulenten Ausbreitungsvorgange.
ZAMM Bd. lY, p. 468, 1926.
82. Schlichting, H. : Ueber das ebene Windschattenproblem. Ing.-Arch.
Bd. I., p. 533, 1930.
83. Swain, L. M. : Proc. Eoy. Soc. London. A. 125, 129, P- 647.
84. Betz, A. : Ein Verfahren zur direkten Ermittlung dea Profilwlderstandes.
Z. f. M. Bd. 16., p. 42, 1925.
85. Jones, B. M. : The Measurement of Profile Drag by the Pitot Transverse
Method. ARC Eep. I6888, I936.
86. Elias, F. : Die Warmeiibertragung einer geheizten Platte an stromende
Luft. ZAMM, 1929, p. 434.
87. Eeichardt, H. : Die Warmeiibertragung in turbulenten Eeib-'jngsschlchten.
ZAMM 20, 1940, p. 297.
88. Olsaon, Gran. : Geachw.— und Temperattir— Vert ei lung hinter elnem Gltter
bel turbulenter Stromung. ZAMM, I936, p. 257.
89. Schultz-Grunow, F. : Neues Eeibungawiderstandsgesetz fur glatte Platten.
Luftf. Forschung 1940, p. 239.
90 : W^ierstandsmeasungen an rauhen Ereiszylindern.
ARC Eep. 1283, 1929.
91. Forthmann, E. : Ueber turbulente Strahlausbreitung. Ing.-Arch. Bd. V,
p. 42, 1934.
WACA TM No. 1218 . 101
92. Taylor, G. I.: The Transport of Vorticity and Heat Troiogh Fluids in
Turtulent Motion. Proc. Boy. Soc. A, Vol. I35, p. 685, 1932.
93. Fage and Falkner: Note on Experiments on the Temperatiire and Velocity
in the Wake of a Heated Cylindrical Obstacle. Proc. Roy. Soc. A.
Vol.135, P- 702, 1932.
914-. Lorenz, H. : Zeitschr. F. techn. Physik 15, p. 376, 193^4-.
95. Hagen: Pogg. Ann. Bd. k6, p. U23, l839-
96. Froude: Experiments on the Svirface Friction. Brit. Ass. Eep., I872.
97- Eryden, H. L. and Kuethe, A. M. : Effect of TurTDiolence in Windtunnel
Measurements. KACA Eep. Wo. 3l<-2, 1929 .
102 MCA TM No. 12l8
BIBLIOGRAPHJ ^
Origin of Turbulence (chapter XXI)
101. Reynolds, 0.: Sci. papers 2, I883.
102. Eayleigh, Lord: Papers 3, 1887.
103. LorenZj H. A.: Athandlungen u'ber thsoretische Physik. I k3,
Leipzig, 1907.
104. Sommerfeld, A. : Ein Beitrag zur Hydrodynami schen Erklarung der
turbulenten FlussigkeitslDewegung. Atti. d. IV" congr. int. dei
Mathem. Eom I9O9.
105. V. Miees, P.: Beitrag zum Oszlllationstheorem. Helnrich Weter
Festschr. I912. Derselte: Zur Turtulenz— Theorie . Jahresbericht
d. deutsch.. Mathem. Ver. I9I2.
106. Hopf , L. : Der Verlauf klelner Schwingungen in einer Stromung
relbender Flussigkeit. Ann. Phys. kk, p. 1, 191^; aj^i: Zur
Theorie der Turhulenz. Ann. Phys., 59 p. 538, 1919.
107. Prandtl, L. : Bemerkungen liher die Entstehung der Turhulenz.
Zeitschr. angew. Math. u. Mech. 1, k31, 1921.
108. Tietjens, 0.: Beitrage zum Turhulenzprohlem. Diss. Gottingen, 1922,
and Zeitschr. angew. Math. u. Mech 5, 200, 1925.
109. Toll mi en, W. : Ueber die Entstehung der Turbulenz. Nachr. Ges. Wiss.
Gottingen. Math. Phys. Klasse I929, p. 21.
110. Toll Tn ien, W. : Ein allgemeines Kriterium der Instabilitat laminarer
G-eschwindigkeitsverteilungen. Nachr. Ges. "Wiss. Gottingen. Math.
Phys, Klasse Bd. I, Nr. 5, 1935.
111. Prandtl, L.: Ueber die Entstehung der Turbulenz. Zeitschr. angew.
Math. u. Mech. 11, k07 , 1931.
112. Schlichting, H. : Zur Entstehung der Turbulenz bei der Piatt enstromung.
Nachr. Ges. Wiss. Gottingen. Math. Phys. Klasse, 1933 j P- I8I.
113. Schlichting, H. : Amplitudenverteilung und Energiebilanz kleiner
Storungen bei der Plattenstromung . Nachr. Ges. Wiss, Gottingen.
Math. Phys. Klasse 1935, P. ^8.
WACA TM Wo. 1218 103
Hi*-. Schlichtingj H. : Berechnung der kritischen EeynoldsBchen Zalil elner
EelTDungBBchicht in beschleunigter mid verzogerter Stromimg.
Jahrlsuch 19^0 der deutschen Luftfahrt— Forschung, p. I 97-
115. Schlichtlng, H. ajid Ulrich, A.: Zur Berechnung des Umschlagea
laminar /turtulent. (Preisausschrei'ben 19I1O der Lilienthal
Gtesellscliaft). Lilienthal-Gesellschaft Bericht S. 10, 19i4-l.
ioi+
NACA TM No. 12l8
TABLE Vn . - THE BRAG LAM OF THE 3dOOTH PLATE
I
'^f
0.074
Eel/5
la
^f
0.074 1700
Eel/5 «-
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2.58
(log Ee)
o.4?5
TJ 1
Be =
f „ „ ,2.58 Ee
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0.427
f 2 64
(- 0.407 + log Ee) ■
I
la
U
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HI
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Cf X 103
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7.40
7.13
7.63
2 X IC?
6.43
6.11
6.50
3 X IC?
5.93
5.62
5.85
4 X IC?
5.60
5.33
5.50
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5.37
5.06
5.23
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5.18
2.35
4.92
2.17
5.06
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4.88
2.76
4.62
2.50
4.74
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4.67
2.97
4.46
2.76
4.51
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4.07
3.22
3.96
3.11
3.95
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3.74
3.17
3.67
3.10
3.68
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3.54
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3.50
3.07
3.50
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3.38
3.04
3.40
3.06
3.33
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3.26
2.98
3.28
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3.21
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3.08
2.87
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2.88
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2.77
2.99
2.82
2.94
2 X 10'^
2.56
2.47
2.67
2.58
2.62
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2.16
2.38
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108
1.85
1.83
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2.12
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1-93
1.92
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1.70
1.70
1.61
109
1.56
1.56
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1.43
1.43
1.33
5 X 10^
1.30
1.30
1.19
10^°
1.20
1.20
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105
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106
KACA TM Wo. 1218
Laminar
/////////^yy//y////////yy
Figure 71.-
Turbulent
Laminar and turbulent velocity distribution
in pipe.
Velocity t/Z'^y
r
iJL
i
Time t
Figure 72.- Fluctuation with time of the velocity of
turbulent flow at a fixed position.
WACA TM No. 1218
107
Velocity y
1 Cy
^
Re
crit
***-^
Figure 73.- Drag and drag coefficient of a sphere.
Subcritical
Supercritical
Figure 74.- Flow around a sphere; subcritical
and supercritical (schematic).
108
NACA TM No. 12l8
0,
'77yy777yvZ7y77:^pV7y77777777777'
laminar i turbulent
^crit.
Figure 75.-
Laminar and turbulent boundary layer on a flat plate in
longitudinal flow.
y
uM
V \ \v'
7 '■'■'
y
Figure 76.- Transfer of momentum by the
turbulent fluctuation velocity.
y
u(y,-U_
ti(y)
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Figure 77,- Explanation of the mixing length.
NACA TM No. 12l8
109
to
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NACA TM Ho. 1218
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111
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112
NACA TM No. I2l8
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115
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Figure 85.- Measurement of the drag of an arbitrary roughness.
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116
MCA TM No. 1218
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119
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Figure 90.- Drag of the circular cylinder for various relative roughnesses.
Figure 91.- Velocity profiles in the turbulent friction layer with pressure
decrease and pressure increase (according to Gruschwitz [78 j).
120
WACA TM Wo. 1218
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WACA TM No. 1218
121
01 OA 0.6 0.6 W
Laminar separaiion point
0.2 0.k Yd W~1.0
Figure 93.- Result of the calculation of the turbulent friction layer
according to Gruschwitz, Example profile J 015; Ca_ = 0.
122
NACA TM No. 12l8
Figure 94.- Free turbulence: free jet and wake.
Figure 95.- Plane wake flow; explanatory sketch.
NACA TM No. 12l8
123
>o
<
■0-
-0
V
«o
\
\,
c>
^Is.*
N
\
c:
H
i all
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J
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ci
0)
a
0)
a
o
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0)
CO
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v
bO
12 J+
NACA TM No. 12l8
n^fii
*-x
^-?2
Figure 97,- The free jet boundary; explanatory sketch.
/
u
/
/
/
/
1
i
1.0
0.&
as
OA
o.z
-2 -/
n
* i
^ =
r =
^
Figure 98.- Free jet boundary; distribution of the longitudinal and
transverse velocity.
NACA TM No. 12l8
125
0:^?^
o
.1-1
-t->
cd
-3
o
.—I
Qi
o
a
m
o
n
o
w
U
o
o
0)
«(-l
i
r-l
a
(U
03
as
0)
126
KACA TM no. 1218
^o;Po,g^
^'■^i.Pi.Qo
^ ■■^z^Pz^92
Figure 100,- Determination of the profile drag according to Betz,
Po^^^o
^■P2,92
I'Pi = Po
Figure 101.- Determination of the profile drag according to Jones.
RACA TM Re. 12l8
127
■^^o
v/y/.
Figure 102,- The stability investigation of the Couette flow.
OLS
U[^) Given
Stable J c/<Z?
Neutral ,• c; -0
UmS
■crit
Figure 103.- The neutral stability curve as result of a stability-
investigation (schematic).
U(y) 1 ^m
-3
//?////////// ^y
UMi
Um .
_,
:J
»
7777—
yT.
Figure 104,- Basic flow without and with inflection point.
128
NACA TM Wo. 1218
Staole
Profile with inflection point
^ /Profile without inflection point
^®crit. ^^crit.
•- /fe=
Um^
Figure 105.- Neutral stability curves for velocity profiles without and
with inflection point (schematic).
P/q
D = Pressure side
5 = Suction side
(J = Transition point
Figure 106.- Wing profile with pressure distribution.
WACA TM No. 1218
129
Figure 107.- Approximation of a velocity-
profile by a polygon.
]fi7^dy-0.mUm<^
X = /.356
= 893
7f!r I 1 1 I 1 1 I 1 1 1 1 1 'rt-r
Figure 108.- Streamline pattern and velocity distribution of the plate
boundary layer for neutrally stable disturbance U(y) - basic flow;
u(y) = U(y) + u'(x,y,t) = disturbed velocity distribution;
A = 27r/a = wave length of the disturbance.
130.
NACA TM No. 12l8
fW<fy'iitW(i,
£ '
Dt
^¥7 .^^^-0,itl9
1. Transfer of aaln
to secondary
■otlon
2. Diss Ipatlon
Von Haupt
• ore «ent
as 1J0
Basic flow
TTTT^fT// In n n
-I
S)^Mt2
tttPu; iJn /'
-us Q.ZS «f -^ -2 O 2 -^
Disturbance amplitude Energy balance
Parameters of the disturbance:
V/ave length x = 13.5 6; a6 = 0.466
Velocity of wave propagation c = 0.35 U
m
Angular frequency /3
277
T
0.163
m
UmS"
893
Figure 109.- Neutral disturbance for the friction layer on the flat plate.
Amplitude of the disturbance velocity:
u'(x,y,t) = u-^(y) cos (x - P^t) - U2(y) sin (x - p^t).
WACA TM No. 1218
131
/;/'^if dhL
Urn dy'
0.5 1.0 ulUm
S iO* Z I 5 1 10^
Figure 110.- Neutral stability curve for the friction layer on the flat
plate in longitudinal flow.
132
NACA TM No. 12l8
1
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133
/
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/
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7
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/
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1
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in^ -
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col
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7
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y
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Figure 112.-
-8 -5-4-2 Z it 6
The critical Reynolds number (Uj^^s
A )^^^|. as a function
of A
P4-
13^
NACA TM Wo. 1218
Figure 113.- Stability calculation for the elliptic cylinder of axis
ratio a^A^i = 4,
If).
1.0
Crit. 0.8
0.6
O.h
02
\
-B
^ojzj.
,
\
Flat plate
\
^"-^
i^
\
— -
^.
\
— _
• —
\'
V
^^,
s
V
— ^
**
._
— —
#2 5 ^o^Z 5 l[f Z 5 1QTZ 5 iO^
Figure 114.- Result of the stability calculation for the elliptic cylinder
of axis ratio a-i /b-]^ = 4.
WACA TM No. 1218
135
u
%-'
7 10^ 10^ 10^
Uo
Uo
Vi
1Q^ 10^
Re= 1(^10' ^-
Uo
Uo
8
Flat plate
Uo-t.
A = laminar separation point (independent of Rg - -^)
M = maximum velocity of the potential flow.
Figure 115.- The position of the instability point as a function of the
Reynolds number for the elliptic cylinders of axis ratios
a;^/b^ = 1, 2,4,8.
136
NACA TM No. 12l8
Symmetrical Joukowsky profile: - = 0.15
Uo
h
1
- f
„ L)/f«!J2L
10'
R
^^^yHiT ^—
'^N,,,,Af
'° «^w^~;r" '
ib^
H
Ca=0
Uo
=0,25
.inB^o" t R
10^ !£i
Uo
Uo
Uo
Flat plate
Uo
Re-<lf Wf
2-10^
— t—
U„t,
A = laminar separation point (independent of Re=-^ )
M = maximum velocity of the potential flow
S = stagnation point
Figure 116.- The position of the instability point as a function of the
Reynolds number for a Joukowsky profile for lift coefficients of
Ca = to c^ = 1.
NACA-Lanelev - 9-15-49 - 350
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